Differentiation – Product Rule

The product rule is used to find derivatives when two functions are multiplied together. The key idea is to take the first function times the derivative of the second, then add the derivative of the first times the second. These problems include polynomial, radical, trigonometric, exponential, logarithmic, and table-based product rule examples.

Notes

Notes for Product Rule of Derivatives

Practice Problems & Videos

Use the product rule to find the derivative of the following functions.

\(\textbf{1)}\) \(f(x)=5x^{3} \cos x\)Link to Youtube Video Solving Question Number 1

 

\(\textbf{2)}\) \(f(x)=\sqrt{x} (x^{2}+5)\)Link to Youtube Video Solving Question Number 2

 

\(\textbf{3)}\) \(f(x)=(x^{3}+2)(x^{2}-1)\)Link to Youtube Video Solving Question Number 3

 

\(\textbf{4)}\) \(f(x)=(x^{4}+4x)(x^{2}-2)\)

 

\(\textbf{5)}\) \(f(x)=3e^{x}\sqrt{x}\)

 

\(\textbf{6)}\) Find \(f'(3)\) where \(f(x)=e^{x} g(x),\) \(\,g(3)=6\) and \(\,g'(3)=2\)

 

\(\textbf{7)}\) \(f(x)=x \ln⁡x\)Link to Youtube Video Solving Question Number 7

 

\(\textbf{8)}\) \(f(x)=(2x^{3}+3)(x^{2}-2x+1)\)

 

\(\textbf{9)}\) \(f(x)=x^{2}\sin x\)

 

\(\textbf{10)}\) \(f(x)=x^{4}e^x\)

 

\(\textbf{11)}\) \(f(x)=\left(x^2+1\right)\ln x\)

 

\(\textbf{12)}\) \(f(x)=\left(3x-1\right)\cos x\)

 

\(\textbf{13)}\) \(f(x)=\left(x^2+4\right)\left(2x^3-1\right)\)

 

\(\textbf{14)}\) \(f(x)=\left(x^2+3x\right)e^{2x}\)

 

\(\textbf{15)}\) \(f(x)=\sqrt{x}\ln x\)

 

\(\textbf{16)}\) \(f(x)=\left(x^2-5\right)\sin x\)

 

\(\textbf{17)}\) Find \(h'(2)\) where \(h(x)=f(x)g(x)\), \(f(2)=4\), \(f'(2)=7\), \(g(2)=3\), and \(g'(2)=-1\)

 

\(\textbf{18)}\) \(f(x)=\left(4x^3-2x\right)\left(x^2+6\right)\)

 

\(\textbf{19)}\) \(f(x)=x^2\arctan x\)

 

\(\textbf{20)}\) \(f(x)=\left(x^3+1\right)\left(e^x+\ln x\right)\)

 

 

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of Graphs of Derivatives
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

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