Differentiation – Power Rule

The power rule is one of the most important derivative rules in calculus. It says that when differentiating a power of \(x\), you bring the exponent down in front and subtract 1 from the exponent. These problems include whole-number exponents, negative exponents, fractional exponents, radicals, constants, and expressions that need to be rewritten before using the rule.

Notes

Notes for Power Rule of Derivatives

 

Questions

Find the derivative

\(\textbf{1)}\) \(f(x)=\displaystyle\frac{1}{x^4}\) Link to Youtube Video Solving Question Number 1

 

\(\textbf{2)}\) \(f(x)=x^5\)

 

\(\textbf{3)}\) \(f(x)=x\)

 

\(\textbf{4)}\) \(f(x)=\displaystyle\frac{3}{x}\)Link to Youtube Video Solving Question Number 4

 

\(\textbf{5)}\) \(f(x)=\displaystyle\frac{1}{x}\)

 

\(\textbf{6)}\) \(f(x)=3x^4-2x+8\)Link to Youtube Video Solving Question Number 6

 

\(\textbf{7)}\) \(f(x)=\displaystyle\frac{1}{6}x^3\)Link to Youtube Video Solving Question Number 7

 

\(\textbf{8)}\) \(f(x)=2^{30}\)

 

\(\textbf{9)}\) \(f(x)=x^{30}\)

 

\(\textbf{10)}\) \(f(x)=5\sqrt{x^5}+\frac{10}{x}-x^{300}\)

 

\(\textbf{11)}\) \(f(x)=\sqrt{x}\)

 

\(\textbf{12)}\) \(f(x)=\displaystyle\frac{7}{x^4}\)

 

\(\textbf{13)}\) \(f(x)=x^{2/3}\)

 

\(\textbf{14)}\) \(f(x)=4x^{-5}-3x^{1/2}+9\)

 

\(\textbf{15)}\) \(f(x)=\displaystyle\frac{5}{2}x^{7/3}\)

 

\(\textbf{16)}\) \(f(x)=3\sqrt[4]{x^{9}}\)

 

\(\textbf{17)}\) \(f(x)=x^{-1/3}-\displaystyle\frac{2}{\sqrt{x}}\)

 

\(\textbf{18)}\) \(f(x)=\displaystyle\frac{1}{5}x^5+\frac{1}{3}x^3+\frac{1}{2}x^{-2}\)

 

\(\textbf{19)}\) \(f(x)=\displaystyle\frac{6}{\sqrt[3]{x^2}}\)

 

\(\textbf{20)}\) \(f(x)=x^{12}+\displaystyle\frac{1}{x^3}+\sqrt[4]{x}-\displaystyle\frac{1}{\sqrt{x^7}}+9\)

 

 

See Related Pages\(\)

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