Derivative of Composite Function/Examples/sin(x^2)

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\map \sin {x^2} } = 2 x \map \cos {x^2}$

Proof

Let $y = x^2$.

Let $z = \sin y$.

Then we have:

$z = \map \sin {x^2}$

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds \cos y \cdot 2 x\)

Derivative of Sine Function, Derivative of Square Function

\(\ds \)

\(=\)

\(\ds 2 x \map \cos {x^2}\)

simplification

$\blacksquare$

Sources

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