Derivative of Composite Function/Examples/Square of Cosine of a x + b
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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 $\text {IX}$: $10$.