Derivative of Composite Function/Examples/Reciprocal of Arctangent of x

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 $\text {IX}$: $12$.