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Wize University Calculus 1 Textbook > Derivatives

The Power Rule (Derivative of Polynomials)

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The Power Rule

The first and most common differentiation shortcut is the Power Rule. This rule lets us easily take derivatives of xx to a power without using the limit definition of the derivative.

Power Rule

The derivative of a function of the form f(x)=xn{f(x)=x^{n}} , where nn is a real number, is:
f′(x)=nxn−1\boxed{\quad f'(x)=nx^{n-1}\quad }

Wize Tip
The Power Rule tells us the derivative of any constant function is 0!

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Example: The Power Rule

Find the derivative of the following functions

f(x)=3x3f(x)=3x^3
f′(x)=3×3x3−1=9x2f'(x)=3\times3x^{3-1}=\boxed{9x^2}


g(x)=1x\displaystyle g\left(x\right)=\frac{1}{x}
g(x)=1x=x−1  ⟹  g′(x)=−1x−2=−1x2\displaystyle g\left(x\right)=\frac{1}{x}=x^{-1}\implies\displaystyle g'\left(x\right)=-1x^{-2}=\boxed{\frac{-1}{x^2}}


h(x)=xh(x)=\sqrt x
h(x)=x=x12  ⟹  h′(x)=12x−12=12xh(x)=\sqrt x=x^{\frac{1}{2}}\implies h'(x)=\frac{1}{2}x^{\frac{-1}{2}}=\boxed{\frac{1}{2\sqrt{x}}}


k(x)=2x3\displaystyle k(x)=\frac{2}{\sqrt{x^3}}
k(x)=2x3=2x−32  ⟹  k′(x)=2∗−32x−52=−3x5\displaystyle k(x)=\frac{2}{\sqrt{x^3}}=2x^{\frac{-3}{2}} \implies \displaystyle k'(x)=2*\frac{-3}{2}x^{\frac{-5}{2}}= \boxed{\frac{-3}{\sqrt{x^5}}}
Find ddx(sin⁡2x+π+cos⁡2x+57) \displaystyle\frac{d}{dx}(\sin^2x+\sqrt{\pi}+\cos^2x+5^7)



Extra Practice
Higher Order Derivatives: The Power Rule
If f(x)=x10f\left(x\right)=x^{10} find f(9)(x)f^{\left(9\right)}\left(x\right).
(i.e. the 9th derivative)
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
The Power Rule: Derivatives of Polynomials
Find the derivative of f(x)=x3xf\left(x\right)=\dfrac{x^3}{\sqrt{x}}
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
Higher Order Derivatives: The Power Rule
If f(x)=x10f\left(x\right)=x^{10} find f(9)(x)f^{\left(9\right)}\left(x\right).
(i.e. the 9th derivative)
Find the derivative of f(x)=x3xf\left(x\right)=\frac{x^3}{\sqrt{x}}
Derivative of Polynomials
Suppose f is a function which satisfies f′(0)=π and f(x+y)=f(x)+f(y)f'(0)=\pi \text{ and }f(x+y)=f(x)+f(y) for all x and y. True or false: f(x)=πxf(x)=\pi x satisfies this function.
The Power Rule: Derivatives of Polynomials
Find the derivative of f(x)=x3xf\left(x\right)=\dfrac{x^3}{\sqrt{x}}
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
If 𝑓(𝑥) = 𝑥10, find 𝑓(9)(𝑥).
(i.e. the 9th derivative)
Find the derivative of f(x)=x3xf\left(x\right)=\frac{x^3}{\sqrt{x}}