The derivative of the natural log, $\ln(x)$ is

Derivative of ln(x)

\[\bbox[yellow,5px]{\dfrac{d}{dx}\ln x = \dfrac{1}{x} }\]

Applying the Chain rule, we have

\[\dfrac{d}{dx}\ln(\text{stuff}) = \dfrac{1}{(\text{stuff})} \cdot \dfrac{d}{dx} (\text{stuff}) \]

You’ll usually see this written as

\[\dfrac{d}{dx}\ln u = \dfrac{1}{u} \cdot \dfrac{du}{dx} \]

Practice problems are of course below!

Problem #1

Differentiate $f(x) = \ln (x^2 + 3x -1).$

Problem #2

Differentiate $f(x) = \sqrt{\ln x^2}.$

Problem #3

Differentiate $f(x) = \ln(\ln x).$

Problem #4

Differentiate $f(x) = \ln(x^2 e^x).$

Problem #5

Differentiate $f(x) = \ln\left(\dfrac{\cos x}{5x} \right).$

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