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