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You CAN Ace Calculus

17calculus > derivatives > product rule

Topics You Need To Understand For This Page

basic derivative rules

power rule

Some of the practice problems require you to know the following rules also (in their basic form, not including the chain rule). If you don't know one or more of these rules, no worries. You can filter them from the list of practice problems.

exponential derivative

derivatives of trig functions

You do NOT need to know the chain rule for anything on this page, including practice problems. For practice problems using the product rule and chain rule, see the chain rule page.

Calculus Main Topics

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ATTENTION INSTRUCTORS: The new 2018 version of 17calculus will include changes to the practice problem numbering system. If you would like advance information to help you prepare for spring semester, send us an email at 2018info at 17calculus.com.

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Product Rule

The Product Rule is pretty straight-forward. If you have a function with two main parts that are multiplied together, for example \(h(x)=f(x) \cdot g(x)\), the derivative is

\(\displaystyle{ \frac{dh}{dx} = }\) \(\displaystyle{ \frac{d}{dx}[f \cdot g] = }\) \(\displaystyle{ f'\cdot g + f \cdot g' }\)

An interesting thing to notice about the product rule is that the constant multiple rule is just a special case of the product rule. For example, if you have function \(f(x) = cg(x)\), the product rule says \(f'(x) = (c)' g(x) + c g'(x) =\) \( 0 + c g'(x) = c g'(x)\). Notice that we use the constant rule to say that \(d[c]/dx = 0\).

And that's all you need to know to use the product rule. A common mistake many students make is to think that the product rule allows you to take the derivative of both terms and multiply them together. WRONG! If this confuses you, go back to the top of the page and reread the product rule and then go through some examples in your textbook.

Before you start using the product rule, it is important to know where it comes from. So take a few minutes to watch this video showing the proof of the product rule.

PatrickJMT - Product Rule Proof

Okay, practice problem time.
Once you are finished with those, the quotient rule is the next logical step.

 
quotient rule →

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practice filters

use basic derivatives and product rule only (11)

use trig rules (4)

use exponential and/or logarithmic rules (4)

Practice Problems

Instructions - - Unless otherwise instructed, calculate the derivatives of the following functions using the product rule, giving your final answers in simplified, factored form.

Level A - Basic

Practice A01

\(f(x)=x(25-x)\)

solution

Practice A02

\(f(x)=(x-2)(x+3)\)

answer

solution

Practice A03

\(f(x)=(2x+3)(3x-2)\)

solution

Practice A04

\(h(x)=(x^2+5x+7)(x^3+2x-4)\)

answer

solution

Practice A05

\(g(x)=(x^3-7x^2+4)(3x^2+14)\)

answer

solution

Practice A06

\(g(x)=(3x^4+2x-1)(x^5-2x^2)\)

solution

Practice A07

\(\displaystyle{f(x)=x^3e^x}\)

answer

solution

Practice A08

\(\displaystyle{y=7e^{2x}}\)

answer

solution

Practice A09

\(\displaystyle{e^x\sqrt{x}}\)

answer

solution

Practice A10

\(\displaystyle{x^4\tan(x)}\)

answer

solution

Practice A11

\(\displaystyle{f(x)=\cos(x)\sin(x)}\)

answer

solution


Level B - Intermediate

Practice B01

\(\displaystyle{g(x)=(x^2+3x-5)\sqrt[3]{x}}\)

answer

solution

Practice B02

\(\displaystyle{f(x)=\left[\frac{1}{x^2}-\frac{3}{x^4}\right](x+5x^3)}\)

answer

solution

Practice B03

\(\displaystyle{H(u)=(u-\sqrt{u})(u+\sqrt{u})}\)

solution

Practice B04

\(f(x)=x(2x-1)(2x+1)\)

solution

Practice B05

\(f(x)=(2x+1)(4-x^2)(1+x^2)\)

solution

Practice B06

\(\displaystyle{\frac{3}{x}\cot(x)}\)

answer

solution


Level C - Advanced

Practice C01

\(\displaystyle{f(x)=4x^2e^x\sin(x)\sec(x)}\)

solution

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