Derivative of Composite Function/Examples/Logarithm of 1 + x^3

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\map \ln {1 + x^3} } = \dfrac {3 x^2} {1 + x^3}$

Proof

Let $u = 1 + x^3$.

Let $y = \ln u$.

Thus we have:

$y = \map \ln {1 + x^3}$

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds \dfrac 1 u \cdot 3 x^2\)

Power Rule for Derivatives, Derivative of Natural Logarithm

\(\ds \)

\(=\)

\(\ds \dfrac {3 x^2} {1 + x^3}\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $14$.