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

Flashcards

Derivative of x^n

The derivative of x raised to the power of n is n multiplied by x raised to the power of n minus 1.

Derivative of e^x

The derivative of e raised to the power of x is simply e raised to the power of x.

Derivative of a^x

The derivative of a raised to the power of x is a raised to the power of x multiplied by the natural logarithm of a.

Derivative of ln(x)

The derivative of the natural logarithm of x is 1 divided by x.

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Derivative of sin(x)

The derivative of the sine of x is the cosine of x.

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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))