Calculus
Textbooks
Boundless Calculus
Inverse Functions and Advanced Integration
Calculus Textbooks Boundless Calculus Inverse Functions and Advanced Integration
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus

Section 1

Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions

Book Version 1
By Boundless
Boundless Calculus
Calculus
by Boundless
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11 concepts
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Inverse Functions

An inverse function is a function that undoes another function.

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Derivatives of Exponential Functions

The derivative of the exponential function is equal to the value of the function.

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Logarithmic Functions

The logarithm of a number is the exponent by which another fixed value must be raised to produce that number.

Derivatives of Logarithmic Functions

The general form of the derivative of a logarithmic function is $\frac{d}{dx}\log_{b}(x) = \frac{1}{xln(b)}$.

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The Natural Logarithmic Function: Differentiation and Integration

Differentiation and integration of natural logarithms is based on the property $\frac{d}{dx}\ln(x) = \frac{1}{x}$.

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The Natural Exponential Function: Differentiation and Integration

The derivative of the exponential function $\frac{d}{dx}a^x = \ln(a)a^{x}$.

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Exponential Growth and Decay

Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value.

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Inverse Trigonometric Functions: Differentiation and Integration

It is useful to know the derivatives and antiderivatives of the inverse trigonometric functions.

Hyperbolic Functions

$\sinh$ and $\cosh$ are basic hyperbolic functions; $\sinh$ is defined as the following: $\sinh (x) = \frac{e^x - e^{-x}}{2}$.

Indeterminate Forms and L'Hôpital's Rule

Indeterminate forms like $\frac{0}{0}$ have no definite value; however, when a limit is indeterminate, l'Hôpital's rule can often be used to evaluate it.

Bases Other than e and their Applications

Among all choices for the base $b$, particularly common values for logarithms are $e$, $2$, and $10$.

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