## Equation of a Tangent Line: Problems and Solutions

### Equation of a Tangent Line or Normal Line: Problems and Solutions

Are you working to find the equation of a tangent line (or normal line) in Calculus? Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself.

To answer these questions, you will almost always use the Point-Slope form of a line. Recall that if a line has slope *m *and contains the point $(x_0, y_0)$, then you can write its equation as:

**Point-Slope form of a line:**

$$\bbox[yellow,5px]{y – y_0 = m(x – x_0)}$$

The problem statement typically specifies the point $(x_0, y_0)$, and so really these problems come down to determining the slope *m *of the line — which we’ll address below.

You will use that equation again and again; memorize it if you don’t know it already.

(It’s just a variant on the definition of slope: $m = \dfrac{y – y_0}{x – x_0}.)$

**I. Tangent Line to a Curve**

*Very* frequently in beginning Calculus you will be asked to find an equation for the line **tangent** to a curve at a particular point. We’re calling that point $(x_0, y_0)$.

To find the line’s equation, you just need to remember that the tangent line to the curve has slope equal to the derivative of the function evaluated at the point of interest:

$$\bbox[yellow,5px]{m_\text{tangent line} = f'(x_0)}$$

That is, find the derivative of the function $f'(x)$, and then evaluate it at $x = x_0$. That value, $f'(x_0),$ *is *the slope of the tangent line.

Hence we can write the equation for the tangent line at $(x_0, y_0)$ as

\[\bbox[10px,border:2px solid blue]{

\begin{align*}

y – y_0 &= m_\text{tangent line}(x – x_0) \\[8px]
y – y_0 &= f'(x_0)(x – x_0)

\end{align*} } \]

If those equations look abstract to you, don’t worry. As soon as you work a few problems, the process will make sense — we promise.

**II. Normal Line to a Curve**

Sometimes instead a question will ask you instead to find the line **normal** to a curve. That’s the same thing as asking for the line that is **perpendicular** to the curve.

You will again use the Point-Slope form of a line. But now to compute the slope of the line, recall that the slopes of perpendicular lines are the negative reciprocals of each other ($m_2 = -\dfrac{1}{m_1}$). We want the slope of the line that is perpendicular to the curve at a point, and hence that is perpendicular to the tangent line to the curve at that point:

\[\bbox[yellow,5px]{

\begin{align*}

m_\text{normal line} &= \frac{-1}{m_\text{tangent line}}\\[12px]
&= \frac{-1}{f'(x_0)}

\end{align*}}\]

Hence we can write the equation for the normal line at $(x_0, y_0)$ as

\[\bbox[10px,border:2px solid blue]{

\begin{align*}

y – y_0 &= m_\text{normal line}(x – x_0) \\[8px]
y – y_0 &= \frac{-1}{f'(x_0)}(x – x_0)

\end{align*} } \]

We recommend *not* trying to memorize all of the formulas above. Instead, remember the Point-Slope form of a line, and then use what you know about the derivative telling you the slope of the tangent line at a given point. The problems below illustrate.

Problem 1 illustrates the process of putting together different pieces of information to find the equation of a tangent line.

Problem 2 requires that you find the pieces of information before you can put them together.

Since $g'(5) = 2$, we know that the slope of the tangent line at $x = 5$ is $m=2$. Furthermore, the tangent line contains the point (5, -3), since it passes through (grazes) that point on the curve.

Then using the point-slope form of a line that contains the point $(x_o, y_o)$ we have

\begin{align*}

y – y_o &= m(x – x_o) \\

y – (-3) &= 2(x-5) \\

y + 3 &= 2(x-5) \quad \cmark

\end{align*}

We can stop there, since we have found a valid equation for the line. OR, we can continue and put the equation into slope-intercept form:

\begin{align*}

y+3 &= 2x – 10 \\

y &= 2x – 10 -3 \\

y &= 2x – 13 \quad \cmark

\end{align*}

Either answer is valid and correct.

**(a)**Find the equation to the line tangent to the curve at the point (1, 5).

**(b)**Find the equation of the line normal (perpendicular) to the curve at the point (1, 5).

Hence we just need its slope *m*, which is is the same as the slope of the curve at that point $(x=1).$ And that slope equals the function’s derivative at that point:

\begin{align*}

f(x) &= x^3 – x + 5\\[8px]
f'(x) &= 3x^2 -1 \\[8px]
f'(1) &= 3(1) – 1 = 2

\end{align*}

Hence the tangent line has slope $m_\text{tangent} = f'(1) = 2.$

Then we can write the equation of this tangent line, using the point-slope form of a line that has slope $m_\text{tangent} = 2$ and contains the point $(1, 5)$:

\begin{align*}

y – y_o &= m_\text{tangent}(x – x_o) \\[8px]
y – 5 &= (2)(x – 1) \quad \cmark

\end{align*}

We can stop there, since we have found a valid equation for the line. OR, we can continue and put the equation into slope-intercept form:

\begin{align*}

y – 5 &= 2x -2 \\[8px]
y &= 2x – 2 + 5 \\[8px]
y&= 2x +3 \quad \cmark

\end{align*}

Either answer is valid and correct.

$$m_2 = -\frac{1}{m_1}$$

From part (a) we know that the line tangent to the curve at the point (1,5) has slope $m_\text{tangent} = 2$, and so our perpendicular line at that point has slope

\begin{align*}

m_\text{normal} &= -\frac{1}{m_\text{tangent}} \\[8px] &= -\frac{1}{2}

\end{align*}

Hence the normal line has slope $m_\text{normal} = -\dfrac{1}{2}.$

This line passes through the curve at point (1, 5) and has slope $m_\text{normal} = -\dfrac{1}{2}$, and so its equation is given by

\begin{align*}

y – y_o &= m_\text{normal}(x – x_o) \\[8px]
y-5 &= -\dfrac{1}{2} (x -1) \quad \cmark

\end{align*}

We can stop there, since we have found a valid equation for the line. OR, we can continue and put the equation into slope-intercept form:

\begin{align*}

y – 5 &= -\dfrac{1}{2}x + \dfrac{1}{2} \\[8px]
y &= -\dfrac{1}{2}x + \dfrac{1}{2} + 5 \\[8px]
y &= -\dfrac{1}{2} x + \frac{11}{2} \quad \cmark

\end{align*}

Either answer is valid and correct.

*[This is a more challenging problem, submitted by a student in the comments below.]*

A line normal (perpendicular) to the curve $y = 2x^2$ at a point in the first quadrant also passes through the point $\left(0, \dfrac{3}{4}\right)$. Find an equation for this line.

As usual, a quick figure helps a lot. Here we’ve shown the curve $y = x^2,$ and the line that passes through the point $(0, \frac{3}{4})$. This line intersects the curve at the point we’re calling $(x_1, y_1).$

**Note that naming this point on the curve with coordinates like this is crucial to our solution.**

There are several pieces of information we have to put together to solve this problem.

(1) The first is that the slope of a line that is normal (perpendicular) to this curve at the point $(x_1, y_1)$ is given by

\[m_\text{normal, at $x=x_1$} = -\dfrac{1}{f'(x_1)} \]
Since $f(x) = 2x^2,$ we have $f'(x) = 4x.$

For the particular point of interest, $(x_1, y_1),$ where the line intersects the curve, we can write the slope of the normal line as

\[m_\text{normal, at $x=x_1$} = -\dfrac{1}{4x_1} \quad \triangleleft \quad (1) \]
Keep that in mind for a moment.

(2) We *also* know that the line contains the points $(0, \frac{3}{4})$ and $(x_1, y_1).$ Hence we can write its slope as

\[m_\text{line} = \frac{y_1-\frac{3}{4}}{x_1-0} \quad \triangleleft \quad (2) \]
Now, the magic: the line must actually meet both conditions (1) and (2), and so we must have

\[m_\text{normal, at $x=x_1$} = m_\text{line}\]
This requirement will let us solve for $y_1,$ as you’ll see:

\begin{align*}

m_\text{normal, at $x=x_1$} &= m_\text{line} \\[8px]
-\dfrac{1}{4x_1} &= \frac{y_1-\frac{3}{4}}{x_1} \\[8px]
-\dfrac{1}{4} &= y_1-\frac{3}{4} \\[8px]
– y_1 &= -\frac{3}{4} + \dfrac{1}{4} \\[8px]
-y_1 &= -\frac{1}{2} \\[8px]
y_1 &= \frac{1}{2} \quad \triangleleft

\end{align*}

We now have the *y*-value where the line intersects the curve. That leaves us with a few steps to go, since the question asked for the equation of the line. So let’s next determine $x_1$:

We know that the point $(x_1, y_1)$ lies on the curve $y = 2x^2,$ and so since $y_1 = \frac{1}{2}$ we must have

\begin{align*}

\frac{1}{2} &= 2x_1^2 \\[8px]
x_1^2 &= \frac{1}{4} \\[8px]
x_1 &= \pm \sqrt{\frac{1}{4}} \\[8px]
&= \pm \frac{1}{2}

\end{align*}

Ah, but the problem specifies “at a point in the first quadrant,” and so we choose the *positive* solution:

\[x_1 = \frac{1}{2} \quad \triangleleft \]
And then we immediately know the slope of the normal line, since we decided (1) above

\[m_\text{normal, at $x=x_1$} = -\dfrac{1}{4x_1} \]
we must have

\begin{align*}

m_\text{normal, at $x=1/2$} &= -\dfrac{1}{4\left(\frac{1}{2} \right)} \\[8px]
&= – \frac{1}{2} \quad \triangleleft

\end{align*}

Now we’re actually almost done: we know that the slope of the normal line is $m_\text{normal} = -\dfrac{1}{2},$ and since we were told the line passes through the point $(0, \frac{3}{4})$ we know it has *y*-intercept $b = \frac{3}{4}.$ Hence using the slope-intercept form of a line:

\begin{align*}

y &= m_\text{normal}x + b \\[8px]
y &= -\frac{1}{2}x + \frac{3}{4} \quad \cmark

\end{align*}

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*all*of our Calculus problems and solutions? Buy full access now — it’s quick and easy!

Can someone help me about this.

Find the slope of the tangent line to the graph of the following functions?

f(x) = x² + 4

f(x) = 1 – x³

Thanks for asking. Following the steps above will get you the answer you’re after!

We’ll address just your first function, \(f(x) = x^2 + 4\), since you can apply to same approach to any function.

Problems like this always specify an x-value for where the line is tangent to the curve. Since you didn’t specify what your problem asks, we’re going to call that value x = a. The y-value of that point on the curve is then y = f(a), and so the point is (a, f(a)). For instance, if the problem asks you to find the tangent line at the point x = 2, meaning a = 2, then since it’s on the function’s curve we know its y-value is y = f(2) = 2^2 + 4 = 8, so the point is (2, 8).

We know that the slope of the curve at this point is: slope m = f'(a). Since f'(x) = 2x, the slope is thus m = f'(a) = 2a. For instance, if as above a = 2, then the slope of the curve at this point is m = f'(2) = 2*2 = 4.

This gives us all the information we need: the tangent line has slope m = f'(a) = 2a, and it just touches the curve at the point (a, f(a)). Then using the point-slope form of the line, we can write it as

\[y – f(a) = 2a(x – a)\]

Once again, if for instance a = 2, we have the slope m = f'(2) = 4, and it is tangent to the curve at the point (2, 8). Hence the equation of the line would be

\[y – 8 = 4(x – 2)\]

We hope that helps, and please let us know if you have other questions!

Kindly help me in this one:

Show that the equation of tangent to \(x^2+xy+y=0\) at the point \((x_1,y_1)\) is given as \((2x_1+y_1)x+(x_1+1)y+y_1=0\)

Thanks for asking! For anyone reading this problem later: it requires implicit differentiation, which we don’t currently have anything publicly available for. So if the first part of this solution doesn’t make sense, it will later.

Assuming you

doknow implicit differentiation, then we find the derivative as\[2x + y + x \dfrac{dy}{dx} + \dfrac{dy}{dx} = 0\]

Solving for \(\dfrac{dy}{dx}\) then gives

\[ \dfrac{dy}{dx} = \frac{-(2x+y)}{x+1} \]

Hence the slope of the tangent line at the point \(x_1, y_1)\) is

\[\left. \dfrac{dy}{dx} \right|_{(x_1, y_1)} = \frac{-(2x_1+y_1)}{x_1+1} \]

As described above, we now use the point-slope form to write the equation of the tangent line:

\[y – y_1 = \left. \dfrac{dy}{dx} \right|_{(x_1, y_1)} \left(x – x_1 \right)\]

so

\[y – y_1 = \frac{-(2x_1+y_1)}{x_1+1} \left(x – x_1 \right) \]

From here, it’s “just” algebra to multiply everything out and simplify . . . although there is one trick you’ll need at the very end to get the answer in the form requested: note that from the original equation, we know that

\[ x_1^2 + x_1y_1 + y_1=0\]

That fact lets you simplify the many terms you end up with into the form requested.

We very much hope that helps, and thanks again for writing with a fun problem! : )

for the curve y=x^2 + x at what point does the normal line at (0,0) intersect the tangent line at (1,2)?

Nice question. Thanks for asking!

Our solution will have three big steps: (1) Find the equation of the normal line at (0, 0). (2) Find the equation of the tangent line at (1, 2). (3) Find where those two lines intersect.

(1) As explained above, since we have f'(0) = 1, the normal line has slope m_normal = -1. Hence the equation of this normal line is [y – 0 = -(x -0)], or just y = -x.

(2) We have f'(1) = 3, and so the slope of the tangent line is m_tangent = 3. The equation of this tangent line is thus y -2 = 3(x-1), or y = 3x + 1.

(3) Then setting the equations for those two lines equal to each other, we find x = 1/4. (We’ll leave that algebra to you.)

We hope that all makes sense and is helpful!

F(x)=4x^2-4x+1. find the points of tangent where it is perpendicular to the line 2y+x-4=0

Great question! We’re going to assume that there’s an equals-sign at the end of the line’s equation, so “…perpendicular to the line $2y+x-4=0$.” If that’s not right, we trust you’ll be able to follow the solution steps and adjust accordingly.

Our solution has three big steps: (1) First, we need the slope of that line. (2) Determine the slope of lines perpendicular to that one. (3) Find points on the function’s curve that have tangent lines with that same slope.

Step 1: The line’s slope is easy to see if we rewrite the equation in Point-Slope form:

\begin{align*}

2y+x-4 &= 0 \\[8px]

2y &= -x +4 \\[8px]

y &= -\dfrac{1}{2} x + 2 \quad \blacktriangleleft \\[8px]

\end{align*}

Now that the equation for the line is written in Point-Slope form, $y = mx + b,$ we can readily see the slope is $m_{\text{line}}=-\dfrac{1}{2}$.

Step 2: We want our tangent lines to be perpendicular to this line, so we want them to have slope $m_\text{tangent} = -\dfrac{1}{m_\text{line}}:$

\[ m_{\text{tangent}}= – \dfrac{1}{- \frac{1}{2}} \]

\[ m_{\text{tangent}} = 2 \quad \blacktriangleleft \]

Step 3: Now we need to find where $F(x)$ has slope $m_{\text{tangent}}=2:$ We first need the function’s derivative:

\begin{align*}

F(x) &= 4x^2 – 4x + 1 \\[8px]

F'(x) &= 8x – 4 \quad \blacktriangleleft \\[8px]

\end{align*}

Since $F'(x)$ is the slope of $F(x)$, we set $F'(x)=2$ and solve for $x$.

\begin{align*}

F'(x) &= 8x – 4 \\[8px]

8x-4 &= 2 \\[8px]

8x &= 6 \\[8px]

x &= \dfrac{3}{4} \quad \blacktriangleleft \\[8px]

\end{align*}

That’s the x-value of the one point where the tangent has the slope we’re after. Now let’s find its y-value:

\begin{align*}

F \left( \dfrac{3}{4} \right) &= 4 \left( \dfrac{3}{4} \right)^2 – 4 \left( \dfrac{3}{4} \right) + 1 \\[8px]

&=4 \left( \dfrac{9}{16} \right) – 4 \left( \dfrac{3}{4} \right)+1 \\[8px]

&= \dfrac{9}{4} – 3 + 1 \\[8px]

&= \dfrac{9}{4} – 2 \\[8px]

&= \dfrac{1}{4} \quad \blacktriangleleft \\[8px]

\end{align*}

Finally, the point where $F(x)$ is tangent to the line perpendicular to $2y + x -4 = 0$ is:

\[ ( 3/4, 1/4) \cmark \]

We hope that helps!

Find the equation(s) of any normals to the curveat points where the tangent

y= x + 1/xis parallel to the line

y = -3x.Let’s break this down:

Step 1: We need ” points where the tangent is parallel to the line y = -3x.” Since that line has slope equal to -3, we’re looking for points on the curve where the tangent line has that slope.And since the slope of the tangent line to the curve at the point $(a, f(a))$ has slope $m_\text{tangent} = f'(a),$ we need to identify the points where $f'(a) = -3.$ So let’s find those:

\begin{align*}

f(x) &= x + \frac{1}{x} \\[8px]

f'(x) &= 1 – \frac{1}{x^2} \\[8px]

\end{align*}

That’s true for any point . We want the particular points where $f'(a) = -3$:

\begin{align*}

f'(a) = 1 – \frac{1}{a^2} &= -3 \\[8px]

– \frac{1}{a^2} &= -4 \\[8px]

a^2 &= \frac{1}{4} \\[8px]

a &= \pm \frac{1}{2} \quad \blacktriangleleft

\end{align*}

Hence we have two points where the tangent line has slope equal to -3: $x_1 = -\dfrac{1}{2}$ and $x_2 = \dfrac{1}{2}$. See the top graph below.

Next, let’s find the equations of the lines that are normal (perpendicular) to the curve at these two points. To write the equation of a line, we need to know its slope, and one point on the line.

Now we already know the slope of these lines:

\[m_\text{normal} = -\frac{1}{m_\text{tangent}} = -\frac{1}{(-3)} = \frac{1}{3} \quad \blacktriangleleft\]

So, we just need both the x-value and y-value of one point on the line. We know from above the x-value of one point on each line, so we just need to find the corresponding y-value of those two points:

We know from above the x-value of one point on each line, so we just need to find the corresponding y-value of those two points:

Recall the original function

\[y = x + \frac{1}{x}\]

Hence for $x_1 = -\dfrac{1}{2}$, we have

\[ y_1 = -\frac{1}{2} + (-2) = -\frac{5}{2} \quad \blacktriangleleft\]

and so the point $(-\dfrac{1}{2}, -\dfrac{5}{2})$ is on the line and its equation, in point-slope form, is

\begin{align*}

y – \left(- \frac{5}{2}\right) &= \frac{1}{3}\left[ x – \left(- \frac{1}{2}\right)\right] \\[8px]

y + \frac{5}{2} &= \frac{1}{3}\left( x + \frac{1}{2}\right) \quad \cmark

\end{align*}

And for $x_2 = \dfrac{1}{2}$ we have

\[y_2 = \frac{1}{2} + 2 = \frac{5}{2} \quad \blacktriangleleft\]

and so the point $(\dfrac{1}{2}, \dfrac{5}{2})$ is on the line and its equation, in point-slope form, is

\[ y – \frac{5}{2} = \frac{1}{3}\left( x – \frac{1}{2}\right) \quad \cmark\]

Our final answer is the two equations with the green checkmarks next to them, and we’ve plotted those two lines, and the orignal function’s curve, on the lower graph below. BUT, much more important is the process we used to find those equations, and we hope that makes sense.

And we hope to have helped! : )