- Differentiate exponential, logarithmic, and trigonometric functions
Derivatives of Exponential, Logarithmic, and Trigonometric Functions
The Natural Exponential Function
The function [latex]f(x) = e^x[/latex] has a unique property: it’s its own derivative.
derivative of [latex]e^x[/latex]
[latex]\dfrac{d}{dx}(e^x) = e^x[/latex]
More generally, using the chain rule:
[latex]\dfrac{d}{dx}(e^{g(x)}) = e^{g(x)} \cdot g'(x)[/latex]
Think of it as: [latex]e[/latex] raised to the power of a function, times the derivative of that function.
Find the derivative of [latex]y=\dfrac{e^{x^2}}{x}[/latex].
General Exponential Functions
derivative of [latex]b^x[/latex]
For exponential functions with base [latex]b \neq e[/latex]:
[latex]\dfrac{d}{dx}(b^x) = b^x \ln b[/latex]
More generally:
[latex]\dfrac{d}{dx}(b^{g(x)}) = b^{g(x)} \cdot g'(x) \cdot \ln b[/latex]
The extra factor of [latex]\ln b[/latex] accounts for the base being different from [latex]e[/latex].
The Natural Logarithm
derivative of [latex]\ln x[/latex]
For [latex]x > 0[/latex]:
[latex]\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}[/latex]
More generally, using the chain rule:
[latex]\dfrac{d}{dx}(\ln(g(x))) = \dfrac{g'(x)}{g(x)}[/latex]
This applies when [latex]g(x) > 0[/latex].
Find the derivative of [latex]f(x)=\ln(x^3+3x-4)[/latex]
General Logarithmic Functions
derivative of [latex]\log_b x[/latex]
For logarithms with base [latex]b \neq e[/latex]:
[latex]\dfrac{d}{dx}(\log_b x) = \dfrac{1}{x \ln b}[/latex]
More generally:
[latex]\dfrac{d}{dx}(\log_b(g(x))) = \dfrac{g'(x)}{g(x) \ln b}[/latex]
Again, the factor of [latex]\ln b[/latex] adjusts for the different base.
Find the derivative of [latex]h(x)= \dfrac{3^x}{3^x+2}[/latex]
Basic Trigonometric Derivatives
derivatives of sine and cosine
[latex]\dfrac{d}{dx}(\sin x) = \cos x[/latex]</center> <center>[latex]\dfrac{d}{dx}(\cos x) = -\sin x[/latex]
Notice the negative sign for cosine—this reflects the relationship between the functions and their graphs.
Watch the signs carefully:
- [latex]\sin x[/latex] differentiates to [latex]+\cos x[/latex]
- [latex]\cos x[/latex] differentiates to [latex]-\sin x[/latex]
At points where [latex]\sin x[/latex] has a horizontal tangent (maximum or minimum), [latex]\cos x = 0[/latex]. Where [latex]\sin x[/latex] is increasing, [latex]\cos x > 0[/latex].
Find the derivative of [latex]f(x)=5x^3 \sin x[/latex].
Find the derivative of [latex]g(x)=\dfrac{\cos x}{4x^2}[/latex].
Other Trigonometric Functions
The remaining trigonometric functions can be differentiated using the quotient rule, since they’re defined as ratios of sine and cosine.
derivatives of all six trigonometric functions
[latex]\dfrac{d}{dx}(\sin x) = \cos x[/latex]
[latex]\dfrac{d}{dx}(\cos x) = -\sin x[/latex]
[latex]\dfrac{d}{dx}(\tan x) = \sec^2 x[/latex]
[latex]\dfrac{d}{dx}(\cot x) = -\csc^2 x[/latex]
[latex]\dfrac{d}{dx}(\sec x) = \sec x \tan x[/latex]
[latex]\dfrac{d}{dx}(\csc x) = -\csc x \cot x[/latex]
Notice the patterns:
- Functions starting with “co-” have negative derivatives
- Squared functions ([latex]\sec^2 x[/latex], [latex]\csc^2 x[/latex]) come from [latex]\tan x[/latex] and [latex]\cot x[/latex]
- Products ([latex]\sec x \tan x[/latex], [latex]\csc x \cot x[/latex]) come from [latex]\sec x[/latex] and [latex]\csc x[/latex]
Find the derivative of [latex]f(x)= \csc x+x \tan x.[/latex]