Derivative of Composite Function/Examples/Logarithm of Cosecant of x

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\ln \cosec x} = -\cot x$

Proof

Let $u = \cosec x$.

Let $y = \ln u$.

Thus we have:

$y = \ln \cosec x$

and so:

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\)

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds \dfrac 1 u \paren {-\cosec x \cot x}\)

Derivative of Cosecant Function, Derivative of Natural Logarithm

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\cosec x} \paren {-\cosec x \cot x}\)

simplification

\(\ds \)

\(=\)

\(\ds -\cot x\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $15$.