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Calculating Derivatives: Problems and Solutions

Calculating Derivatives: Problems and Solutions

Are you working to calculate derivatives in Calculus? Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself.

Jump down this page to: [Power rule, $x^n$] [Exponential, $e^x$] [Trig derivatives] [Product rule] [Quotient rule] [Chain  rule]

CALCULUS SUMMARY: Derivatives and Rules
You can always access our Handy Table of Derivatives and Differentiation Rules via the Key Formulas menu item at the top of every page.
CLICK TO VIEW SUMMARY


I. Power Rule
\[\bbox[yellow,5px]{\dfrac{d}{dx}\left(x^n\right) = nx^{n-1}}\] For example, $\dfrac{d}{dx}\left(x^3\right) = 3x^2;$     $\dfrac{d}{dx}\left(x^{47}\right) = 47x^{46}.$

Two specific cases you’ll quickly remember:
$$\dfrac{d}{dx}\text{(constant)} = 0$$
$$\dfrac{d}{dx}(x) = 1$$

Power Rule Differentiation Problem #1
Differentiate $f(x) = 2\pi$.
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Power Rule Differentiation Problem #2
Find the derivative of $f(x) = \dfrac{2}{3}x^9$.
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Power Rule Differentiation Problem #3
Calculate the derivative of $f(x) = 2x^3 – 4x^2 + x -33$.
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Power Rule Differentiation Problem #4
Differentiate $f(x) = \sqrt{x}$.
Tips iconRecall $\sqrt[n]{x} = x^{1/n}.$
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Power Rule Differentiation Problem #5
Differentiate $f(x) = \dfrac{5}{x^3}$.
Tips iconRecall $\dfrac{1}{x^n} = x^{-n}.$
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Power Rule Differentiation Problem #6
Calculate the derivative of $f(x) = \sqrt[3]{x}\, – \dfrac{1}{\sqrt{x}}$.
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Power Rule Differentiation Problem #7
Find the derivative of $f(x) = \sqrt{x}\left(x^2 – 8 + \dfrac{1}{x} \right)$.
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Power Rule Differentiation Problem #8
Differentiate $f(x) = \left(2x^2 + 1 \right)^2$.
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II. Exponential Function Derivative
$$\bbox[yellow,5px]{\dfrac{d}{dx}e^x = e^x}$$
This one’s easy to remember!
Exponential Differentiation Problem #1
Differentiate $f(x) = e^x + x$.
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Exponential Differentiation Problem #2
Calculate the derivative of $f(x) = e^{1 + x}$.
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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 $f(x) = \sin x – \cos x$.
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Trig Differentiation Problem #2
Calculate the derivative of $f(x) = 5x^3 – \tan x$.
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IV. Product Rule
\[\bbox[yellow,5px]{
\begin{align*}
\dfrac{d}{dx}(fg)&= \left(\dfrac{d}{dx}f \right)g + f\left(\dfrac{d}{dx}g \right)\\[8px] &= [{\small \text{ (deriv of the 1st) } \times \text{ (the 2nd) }}]\, + \,[{\small \text{ (the 1st) } \times \text{ (deriv of the 2nd)}}] \end{align*}}\]
Product Rule Differentiation Problem #1
Differentiate $f(x) = x\sin x.$
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Product Rule Differentiation Problem #2
Calculate the derivative of $f(x) = \left(e^x +1 \right) \tan x.$
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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 Differentiation Problem #1
Differentiate $f(x) = \dfrac{\sin x}{x}.$
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Quotient Rule Differentiation Problem #2
Calculate the derivative of $f(x) = \dfrac{e^x}{x+1}.$
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Quotient Rule Differentiation Problem #3
Find the derivative of $f(x) = \dfrac{3x}{5 – \tan x}.$
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VI. Chain Rule
The Chain Rule is a big topic, so we have a separate page on problems that require the Chain Rule.
Need to use the derivative to find the equation of a tangent line (or the equation of a normal line)? We have a separate page on that topic here.

Have a question, suggestion, or item you’d like us to include? Please let us know in the Comments section below!


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