Derivative of Composite Function/Examples/Reciprocal of Arctangent of x
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Example of Derivative of Composite Function
- $\map {\dfrac \d {\d x} } {\dfrac 1 {\arctan x} } = -\dfrac 1 {\paren {1 + x^2} \arctan^2 x}$
Proof
Let $u = \arctan x$.
Let $y = u^{-1}$.
Thus we have:
- $y = \dfrac 1 {\arctan x}$
and so:
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\) | Derivative of Composite Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {-1} {u^2} \cdot \dfrac 1 {1 + x^2}\) | Power Rule for Derivatives, Derivative of Arctangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 {\paren {1 + x^2} \arctan^2 x}\) | simplification |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $12$.