Derivative of Composite Function/Examples/Square of Cosine of a x + b

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\map {\cos^2} {a x + b} } = -2 a \map \cos {a x + b} \map \sin {a x + b}$

Proof

Let $u = \map \cos {a x + b}$.

Let $y = u^2$.

Thus we have:

$y = \map {\cos^2} {a x + b}$

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds 2 u \paren {-a \map \sin {a x + b} }\)

Power Rule for Derivatives, Derivative of $\map \cos {a x + b}$

\(\ds \)

\(=\)

\(\ds -2 a \map \cos {a x + b} \map \sin {a x + b}\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $10$.