Derivative of Composite Function/Examples/(3x+1)^4

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\paren {3 x + 1}^4} = 12 \paren {3 x + 1}^3$

Proof

Let $u = 3 x + 1$.

Let $y = u^4$.

Then we have:

$y = \paren {3 x + 1}^4$

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds 3 \cdot 4 \paren {3 x + 1}^3\)

Power Rule for Derivatives, Derivative of Identity Function: Corollary

\(\ds \)

\(=\)

\(\ds 12 \paren {3 x + 1}^3\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $3$.