What is the derivative of ln(sin^2(e^x)) ?

The derivative of ln(sin^2(e^x)) is e^xcsc^2(e^x)sin(2e^x)

What is the first derivative of ln(sin^2(e^x)) ?

The first derivative of ln(sin^2(e^x)) is e^xcsc^2(e^x)sin(2e^x)

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derivative of

ln (sin^{2} (e^{x}))

e^{x} csc^{2} (e^{x}) sin (2 e^{x})

Solution steps

\frac{d}{d x} (ln (sin^{2} (e^{x})))

Apply the chain rule:

\frac{1}{sin ^{2} ( e ^{x} )} \frac{d}{d x} (sin^{2} (e^{x}))

= \frac{1}{sin ^{2} ( e ^{x} )} \frac{d}{d x} (sin^{2} (e^{x}))

\frac{d}{d x} (sin^{2} (e^{x})) = e^{x} sin (2 e^{x})

= \frac{1}{sin ^{2} ( e ^{x} )} e^{x} sin (2 e^{x})

Simplify

\frac{1}{sin ^{2} ( e ^{x} )} e^{x} sin (2 e^{x}) : e^{x} csc^{2} (e^{x}) sin (2 e^{x})

= e^{x} csc^{2} (e^{x}) sin (2 e^{x})

What is the derivative of ln(sin^2(e^x)) ?
The derivative of ln(sin^2(e^x)) is e^xcsc^2(e^x)sin(2e^x)
What is the first derivative of ln(sin^2(e^x)) ?
The first derivative of ln(sin^2(e^x)) is e^xcsc^2(e^x)sin(2e^x)