Product Rule Derivatives: Easy, Medium, and Hard
This page walks through some of the examples in the following video found on the JoeCMath YouTube channel. The goal of the video is to walk the viewer through a sequence of eight product rule derivatives that increase in difficulty by utilizing different combinations of the the main derivative rules in association with the product rule.
The Rule
Use the product rule when a function is written as one function multiplied by another function.
\[\frac{d}{dx}\left[f(x)g(x)\right]=f(x)g'(x)+g(x)f'(x)\]A good way to remember the structure is:
\[\text{first}\cdot \text{derivative of second}+\text{second}\cdot \text{derivative of first}\]Easy Example: Product Rule Only
The video starts with:
\[f(x)=x\tan(x)\]Think of this as the product of two functions:
\[f(x)=\underbrace{x}_{\text{first}}\underbrace{\tan(x)}_{\text{second}}\]Their derivatives are:
\[\frac{d}{dx}(x)=1\]and
\[\frac{d}{dx}\left(\tan(x)\right)=\sec^2(x)\]Now plug into the product rule:
\[f'(x)=x\sec^2(x)+\tan(x)\cdot 1\]So:
\[f'(x)=x\sec^2(x)+\tan(x)\]Watch this part: easy example at 0:49.
Medium Examples: Product Rule Plus Another Rule
The medium examples still start with the product rule, but one of the pieces may need another rule after that.
For example:
\[f(x)=x^3e^x\]The two pieces are $x^3$ and $e^x$. Their derivatives are:
\[\frac{d}{dx}(x^3)=3x^2\]and
\[\frac{d}{dx}(e^x)=e^x\]Apply the product rule:
\[f'(x)=x^3e^x+e^x(3x^2)\]You can leave the answer like that, or factor out $e^x$:
\[f'(x)=e^x(x^3+3x^2)\]Watch this part: product rule with the power rule at 2:33.
Another medium-level example in the video is:
\[f(x)=\sin(x)e^{\cos(x)}\]This still uses the product rule first, but the derivative of $e^{\cos(x)}$ needs the chain rule:
\[\frac{d}{dx}\left(e^{\cos(x)}\right)=e^{\cos(x)}(-\sin(x))\]Then:
\[f'(x)=\sin(x)\left[-\sin(x)e^{\cos(x)}\right]+e^{\cos(x)}\cos(x)\]A cleaner factored form is:
\[f'(x)=e^{\cos(x)}\left(\cos(x)-\sin^2(x)\right)\]Watch this part: product rule with the chain rule at 5:21.
Harder Examples: Decide The Outer Rule First
When a problem has products and quotients, choose the outermost rule first.
For example, the video includes:
\[f(x)=\frac{x\sin(x)}{\ln(x)}\]The outer structure is a quotient, so start with the quotient rule. Let:
\[N=x\sin(x)\]and
\[D=\ln(x)\]The numerator $N$ is itself a product, so:
\[N'=x\cos(x)+\sin(x)\]The denominator derivative is:
\[D'=\frac{1}{x}\]Now use the quotient rule:
\[f'(x)=\frac{N'D-ND'}{D^2}\]Substitute back in:
\[f'(x)=\frac{\left(x\cos(x)+\sin(x)\right)\ln(x)-x\sin(x)\left(\frac{1}{x}\right)}{\left(\ln(x)\right)^2}\]Watch this part: quotient rule with product rule inside at 8:44.
The Big Final Pattern
A later example combines the product rule, quotient rule, chain rule, and power rule:
\[f(x)=\frac{e^{\sin(x)}(x^8+x)}{\tan(x)}\]Because the whole expression is a quotient, start by naming the top and bottom:
\[N=e^{\sin(x)}(x^8+x)\]and
\[D=\tan(x)\]The numerator is a product, so find $N’$ with the product rule:
\[N'=e^{\sin(x)}\cos(x)(x^8+x)+e^{\sin(x)}(8x^7+1)\]The denominator derivative is:
\[D'=\sec^2(x)\]Now place those pieces into the quotient rule:
\[f'(x)=\frac{\left[e^{\sin(x)}\cos(x)(x^8+x)+e^{\sin(x)}(8x^7+1)\right]\tan(x)-e^{\sin(x)}(x^8+x)\sec^2(x)}{\tan^2(x)}\]This is the main habit the video builds: do not try to simplify everything mentally at once. Name the pieces, find the needed derivatives, then assemble the rule.
Watch this part: hard example at 26:44.
Timestamp Guide
More Practice
For the full list of timestamped practice problems from this video, use the existing archives: