Derivative of Composite Function/Examples/Reciprocal of (2x+1)^3
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Example of Derivative of Composite Function
- $\map {\dfrac \d {\d x} } {\dfrac 1 {\paren {2 x + 1}^3} } = -\dfrac 6 {\paren {2 x + 1}^4}$
Proof
Let $u = 2 x + 1$.
Let $y = u^{-3}$.
Then we have:
- $y = \paren {2 x + 1}^{-3}$
and so:
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\) | Derivative of Composite Function | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \cdot \paren {-3} \paren {2 x + 1}^{-4}\) | Power Rule for Derivatives, Derivative of Identity Function: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 6 {\paren {2 x + 1}^4}\) | simplification |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $4$.