Are you working to learn implicit differentiation in Calculus? Let’s simplify things and develop a can’t-fail approach, and practice with some typical homework and exam problems — each with a complete solution a click away.

PROBLEM SOLVING STRATEGY: Implicit Differentiation

There are two basic steps to solve implicit differentiation problems:

- Take the derivative $\dfrac{d}{dx}$ of both sides of the equation.
- Use your usual Rules of Differentiation, with one addition: When you take the derivative of a term with a
*y*in it, be sure to multiply by $\dfrac{dy}{dx}$ due to the Chain Rule. - Remember that $\dfrac{d}{dx}\text{(constant)}= 0$ . (It’s a common error to forget that when doing these problems, even among experts.)

- Use your usual Rules of Differentiation, with one addition: When you take the derivative of a term with a
- Solve for $\dfrac{dy}{dx}$.
- Collect all terms with $\dfrac{dy}{dx}$ in them on the left side of the equation, all other terms on the right.
- Factor and divide as necessary to solve for $\dfrac{dy}{dx}$.

*Note*: You may use $y’$ instead of $\dfrac{dy}{dx}$. They are interchangeable:

$$y’ = \dfrac{dy}{dx}$$

Question 1: $x^2 - 2y^2 = 4$

Use implicit differentiation to find $\dfrac{dy}{dx}$ given $x^2 - 2y^2 = 4$.

Question 2: $2\sin(x) \cos(y) = 5$

Use implicit differentiation to find $\dfrac{dy}{dx}$ given $2 \sin x \, \cos y = 0.4$.

Question 3: $e^{(x/y)} = 2x - y$

Use implicit differentiation to find $\dfrac{dy}{dx}$ given $e^{x/y} = 2x - y$.

*Note*: Don't spend time simplifying your final expression. Once you've isolated $\dfrac{dy}{dx}$, stop.

Question 4: $(x+y)^{1/2} + (xy)^{1/2} = 3$

Use implicit differentiation to find $\dfrac{dy}{dx}$ given $\sqrt{x+ y} + \sqrt{xy} = 3$.

*Note: *Don't spend time simplifying your final expression. Once you've isolated $\dfrac{dy}{dx}$, stop.

Question 5: Tangent and normal lines

Consider the relation $y^2x^2 = 256$.**(a)** Find the points where the tangent line to the curve is parallel to the line $y = 3 -4x$.**(b)** Find the points where the tangent line to the curve is perpendicular to the line $y = x$.

Question 6: Second derivative of a circle

Consider the circle $x^2 + y^2 = r^2$, where $r$ is a constant. Find $y^″(x)$.

Question 7: Acutal university exam problem #1

Find the equation of the line tangent to the graph of $x^2y^2 + 2y^3 = 3$ at the point (1,3) on the graph.

Question 8: Actual university exam problem #2

Consider the curve $x + xy + 2y^2 = 6$.**(a)** Find an expression for the slope of the curve at any point $(x, y)$.**(b)** Write an equation for the line tangent to the point (2,1).**(c)** Find the coordinates of all other points on this curve with slope equal to the slope at (2,1).

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