Derivative of Composite Function/Examples/(a x^2 + b x + c)^3

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\paren {a x^2 + b x + c}^3} = 3 \paren {2 a x + b} \paren {a x^2 + b x + c}^2$

Proof

Let $u = a x^2 + b x + c$.

Let $y = u^3$.

Thus we have:

$y = \paren {a x^2 + b x + c}^3$

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds 3 u^2 \paren {\map {\dfrac \d {\d x} } {a x^2} + \map {\dfrac \d {\d x} } {b x} + \map {\dfrac \d {\d x} } c}\)

Power Rule for Derivatives, Sum Rule for Derivatives

\(\ds \)

\(=\)

\(\ds 3 u^2 \paren {2 a x + b}\)

Power Rule for Derivatives, Derivative of Constant

\(\ds \)

\(=\)

\(\ds 3 \paren {2 a x + b} \paren {a x^2 + b x + c}^2\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $29$.