Wize AP Calculus (AB) Textbook > Differentiation 1: Basics & Fundamental Properties

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 xxx 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}}f(x)=xn , where nnn is a real number, is:
f′(x)=nxn−1\boxed{\quad f'(x)=nx^{n-1}\quad }f′(x)=nxn−1​
 

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^3f(x)=3x3

f′(x)=3×3x3−1=9x2f'(x)=3\times3x^{3-1}=\boxed{9x^2}f′(x)=3×3x3−1=9x2​

g(x)=1x\displaystyle g\left(x\right)=\frac{1}{x}g(x)=x1​

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}}g(x)=x1​=x−1⟹g′(x)=−1x−2=x2−1​​

h(x)=xh(x)=\sqrt xh(x)=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}}}h(x)=x​=x21​⟹h′(x)=21​x2−1​=2x​1​​

k(x)=2x3\displaystyle k(x)=\frac{2}{\sqrt{x^3}}k(x)=x3​2​

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}}}k(x)=x3​2​=2x2−3​⟹k′(x)=2∗2−3​x2−5​=x5​−3​​

Find ddx(sin⁡2x+π+cos⁡2x+57)
\displaystyle\frac{d}{dx}(\sin^2x+\sqrt{\pi}+\cos^2x+5^7)dxd​(sin2x+π​+cos2x+57)

Answer

I don't know