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Derivatives of Common Functions

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Questions and Answers

Match the following functions with their derivatives:

$x^n$ = $n x^{n-1}$ $e^x$ = $e^x$ $a^x$ = $a^x ext{ln}(a)$ $ an(x)$ = $ ext{sec}^2(x)$

Match the following inverse trigonometric functions with their derivatives:

$ ext{sin}^{-1}(x)$ = $rac{1}{ ext{sqrt}(1-x^2)}$ $ ext{cos}^{-1}(x)$ = $-rac{1}{ ext{sqrt}(1-x^2)}$ $ ext{tan}^{-1}(x)$ = $rac{1}{1+x^2}$ $ ext{sec}^{-1}(x)$ = $rac{1}{x ext{sqrt}(x^2-1)}$

Match the following logarithmic functions with their derivatives:

$ ext{ln}(x)$ = $rac{1}{x}$ $ ext{log}_a(x)$ = $rac{1}{x ext{ln}(a)}$ $ ext{ln}(a^x)$ = $ ext{ln}(a)$ $ ext{log}_b(b^x)$ = $1$

Match the following trigonometric functions with their derivatives:

<p>$ ext{sin}(x)$ = $ ext{cos}(x)$ $ ext{cos}(x)$ = $- ext{sin}(x)$ $ ext{sec}(x)$ = $ ext{sec}(x) ext{tan}(x)$ $ ext{csc}(x)$ = $- ext{csc}(x) ext{cot}(x)$</p> Signup and view all the answers

Match the following identities with their derivatives:

<p>$rac{d}{dx}[ ext{sin}^{-1}(x)]$ = $rac{1}{ ext{sqrt}(1-x^2)}$ $rac{d}{dx}[ ext{cos}^{-1}(x)]$ = $-rac{1}{ ext{sqrt}(1-x^2)}$ $rac{d}{dx}[ ext{tan}^{-1}(x)]$ = $rac{1}{1+x^2}$ $rac{d}{dx}[ ext{csc}^{-1}(x)]$ = $-rac{1}{x ext{sqrt}(x^2-1)}$</p> Signup and view all the answers

Study Notes

Derivatives of Common Functions

Power Rule: The derivative of xⁿ is nx^n-1
Exponential Function: The derivative of e^x is e^x
Constant Multiple Rule: The derivative of a(x) is a(x) ln(a)
Natural Logarithm: The derivative of ln(x) is 1/x
Logarithm (general base): The derivative of log_a(x) is 1/(x ln(a))
Trigonometric Functions:
- The derivative of sin(x) is cos(x)
- The derivative of cos(x) is -sin(x)
- The derivative of tan(x) is sec²(x)
- The derivative of sec(x) is sec(x) tan(x)
- The derivative of csc(x) is -csc(x) cot(x)
- The derivative of cot(x) is -csc²(x)
Inverse Trigonometric Functions:
- The derivative of sin^-1(x) is 1/√(1-x²)
- The derivative of cos^-1(x) is -1/√(1-x²)
- The derivative of tan^-1(x) is 1/(1+x²)
- The derivative of sec^-1(x) is 1/(|x|√(x²-1))
- The derivative of csc^-1(x) is -1/(|x|√(x²-1))

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Description

Test your knowledge on the derivatives of common functions, including power, exponential, logarithmic, and trigonometric functions. This quiz covers essential rules and formulas you need to solve calculus problems effectively.