Matheno Essentials: Derivatives and Rules Summary

You can always access our Handy Table of Derivatives and Differentiation Rules via the Key Formulas menu item at the top of every page.

I. Power Rule

Most frequently, you will use the Power Rule:
\[\bbox[yellow,5px]{\dfrac{d}{dx}\left(x^n\right) = nx^{n-1}}\] This is just a fancy, compact way of capturing
\begin{align*}
\dfrac{d}{dx}(x) &= 1x^0 = 1 \\
\dfrac{d}{dx}\left(x^2\right) &= 2x\\
\dfrac{d}{dx}\left(x^3 \right) &= 3x^2\\
\dfrac{d}{dx}\left( x^4 \right)&= 4x^3 \\
\vdots\\
\dfrac{d}{dx}\left(x^{157}\right) &= 157x^{156}\\
& \text{etc.}&
\end{align*} The rule works just the same for negative exponents:
\begin{align*}
\dfrac{d}{dx}(x^{-1}) &= -1x^{-2} \\
\dfrac{d}{dx}\left(x^{-2}\right) &= -2x^{-3}\\
\dfrac{d}{dx}\left(x^{-3} \right) &= -3x^{-4}\\
& \text{etc.}&
\end{align*} The rule also captures the fact that the derivative of a constant ($c$) is zero:
$${\bf\text{Constant, c:}} \qquad \dfrac{d}{dx}(c) = \dfrac{d}{dx}cx^0 = 0$$ Finally, because $\sqrt{x}$ comes up so frequently, even though it's easy to compute (as we will below), it's worth memorizing
$${\bf\text{Square root:}} \qquad \dfrac{d}{dx}\sqrt{x} = \dfrac{1}{2\sqrt{x}}$$

Try your hand at using the Power Rule in the following problems. As always, the complete solution is immediately available by clicking "Show/Hide Solution."

Try your hand at using the Power Rule in the following problems. As always, the complete solution is immediately available by clicking "Show/Hide Solution."

Power Rule Differentiation Problem #1

Differentiate $f(x) = 2\pi$.

Power Rule Differentiation Problem #2

Differentiate $f(x) = \dfrac{2}{3}x^9$.

Power Rule Differentiation Problem #3

Differentiate $f(x) = 2x^3 - 4x^2 + x -33$.

Power Rule Differentiation Problem #4

Differentiate $f(x) = x^{1001} + 5x^3 -6x +10,687$.

Power Rule Differentiation Problem #5: Sqrt(x)

Show that $\dfrac{d}{dx}\sqrt{x} = \dfrac{1}{2}\dfrac{1}{\sqrt{x}}$.

Recall that $\sqrt{x} = x^{1/2}.$

Recall that $\sqrt{x} = x^{1/2}.$

Power Rule Differentiation Problem #6

Differentiate $f(x) = \dfrac{5}{x^3}$.

Recall that $\dfrac{1}{x^n} = x^{-n}.$

Recall that $\dfrac{1}{x^n} = x^{-n}.$

Power Rule Differentiation Problem #7

Differentiate $f(x) = \dfrac{1}{x^2} - \dfrac{4}{x^5}$.

Power Rule Differentiation Problem #8

Differentiate $f(x) = \sqrt[3]{x}\, - \dfrac{1}{\sqrt{x}}$.

Power Rule Differentiation Problem #9

Differentiate $f(x) = \sqrt{x}\left(x^2 - 8 + \dfrac{1}{x} \right)$.

Power Rule Differentiation Problem #10

Differentiate $f(x) = \left(2x^2 + 1 \right)^2$.

II. Exponential Function Derivative

Exponential Differentiation Problem #1

Differentiate $f(x) = e^x + x$.

Exponential Differentiation Problem #2

Differentiate $f(x) = e^{1 + x}$.

III. Trig Function Derivatives

\[ \bbox[yellow,5px]{
\begin{align*}
\frac{d}{dx}\left(\sin x\right) &= \cos x &&& \frac{d}{dx}\left(\csc x\right) &= -\csc x \cot x \\ \\
\dfrac{d}{dx}\left(\cos x\right) &= -\sin x &&& \frac{d}{dx}\left(\sec x\right) &= \sec x \tan x \\ \\
\dfrac{d}{dx}\left(\tan x\right) &= \sec^2 x &&& \frac{d}{dx}\left(\cot x\right) &= -\csc^2 x
\end{align*}} \]
Notice that a negative sign appears in the derivatives of the co-functions: cosine, cosecant, and cotangent.

Trig Differentiation Problem #1

Differentiate $f(x) = \sin x - \cos x$.

Trig Differentiation Problem #2

Differentiate $f(x) = 5x^3 - \tan x$.

IV. Product Rule

Product Rule Problem #1

Differentiate $f(x) = x^3 e^x$.

Product Rule Problem #2

Differentiate $f(x) = x\sin x.$

\begin{array}{lll} \text{(A) }\cos x && \text{(B) }\sin x -x\cos x && \text{(C) } \sin x + x\cos x \end{array}

\begin{array}{ll}\text{(D) }-\cos x && \text{(E) None of these} \end{array}

Product Rule Problem #3

Differentiate $g(\theta) = \sin \theta \, \cos \theta$.

Product Rule Problem #4

Differentiate $f(x) = \left(e^x +1 \right) \tan x.$

Product Rule Problem #5

Differentiate $z(x) = x^{5/2} \, e^x \sin x$.

Product Rule Problem #6

Given that $f(2) = 1$, $f'(2) = -3$, $g(2) = 4$, and $g'(2) = 8$, find $(fg)'(2)$.

V. Quotient Rule

\[ \bbox[yellow,5px]{
\begin{align*}
\dfrac{d}{dx}\left(\dfrac{f}{g} \right) &= \dfrac{\left(\dfrac{d}{dx}f \right)g - f\left(\dfrac{d}{dx}g \right)}{g^2} \\[8px]
&=\dfrac{{[{\small \text{(derivative of the numerator) } \times \text{ (the denominator)}]}\\ \quad - \, [{\small \text{ (the numerator) } \times \text{ (derivative of the denominator)}}]}}{{\small \text{all divided by [the denominator, squared]}}}
\end{align*}}\]
Many students remember the quotient rule by thinking of the numerator as "hi," the demoninator as "lo," the derivative as "d," and then singing

"lo d-hi minus hi d-lo over lo-lo"

Quotient Rule Problem #1

Differentiate $f(x) = \dfrac{x^2}{e^x}$.

Quotient Rule Problem #2

Differentiate $f(x) = \dfrac{\sin x}{x}.$

Quotient Rule Problem #3

Differentiate $f(x) = \dfrac{e^x}{x+1}.$

Quotient Rule Problem #4

Differentiate $f(x) = \dfrac{3x}{5 - \tan x}.$

Quotient Rule Problem #5

Differentiate $g(u) = \dfrac{u^3 - 5u^2 + 6}{u -2}$.

Stop after taking the derivatives; don't bother to multiply out the terms and simplify.

Stop after taking the derivatives; don't bother to multiply out the terms and simplify.

Quotient Rule Problem #6

Given that $f(2) = 1$, $f'(2) = -3$, $g(2) = 4$, and $g'(2) = 8$, find $\left(\dfrac{f}{g}\right)'(2)$.

More problems; University Exam Questions

Exam 1 Questions

Find the requested information.**(a)** Let $f(u) = \dfrac{5u - 3}{u^2 + 1}$. Find $f'(u)$.**(b)** Let $g(t) = (t^2 + 3 + t^{-2}) \tan t$. Find $g'(t)$.

VERY Common Exam Question: Find a & b (#1)

Find all $a$ and $b$ such that the function $f(x)$ is differentiable: $$ f(x) = \begin{cases} ax + b & \text{if } x \lt 1 \\ x^4 + x +1 & \text{if } x \geq 1 \end{cases} $$

VERY Common Exam Question: Find a & b (#2)

Find the values of $a$ and $b$ that will make the function $f(x)$ differentiable. $$ f(x) =\begin{cases} ax + b & \text{if } x \lt \pi \\ \sin x & \text{if } x \geq \pi \end{cases} $$

Table of Values

Let $f$ and $g$ be differentiable functions and let the values of $f, g, f'$ and $g'$ at $x=1$ and $x=2$ be given by the table.

\begin{array}{c | c | c | c | c} x & f(x) & g(x) & f'(x) & g'(x)\\ \hline 1 & 5 & 3 & 2 & 7 \\ \hline 2 & -2 & 1 & 4 & 6 \\ \end{array}

Find the numerical value of each of the following:**(a)** the derivative of $\displaystyle{\frac{f}{g}}$ at $x = 1$**(b)** $\displaystyle{\lim_{h \to 0} \frac{9 \cdot f(1+h) - 9 \cdot f(1)}{h}}$**(c)** $\displaystyle{\lim_{h \to 0} \frac{f(2+h) \cdot g(2+h) - f(2) \cdot g(2)}{h}}$**(d)** $\displaystyle{\lim_{h \to 0} \dfrac{g(2) - g(2+h)}{h}}$

\begin{array}{c | c | c | c | c} x & f(x) & g(x) & f'(x) & g'(x)\\ \hline 1 & 5 & 3 & 2 & 7 \\ \hline 2 & -2 & 1 & 4 & 6 \\ \end{array}

Find the numerical value of each of the following:

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Jump down this page to: [Power rule: $x^n$] [Exponential: $e^x$] [Trig derivs] [Product rule] [Quotient rule] [More problems & University exam problems]

[Chain rule (will take you to a new page)]

Update:As of October 2022, we havemuchmore more fully developed materials for you to learn about and practice computing derivatives. Please visit our Calculating Derivatives Chapter toreallyget this material down for yourself. It’s all free, and designed to help you do well in your course.If you just need practice with calculating derivative problems for now, previous students have found what’s below super-helpful. And if you have questions, please ask on our Forum!