derivative of (arctan(x))^2 — step-by-step solution

Let's differentiate:

\frac{d}{d x} ({\arctan (x)}^{2})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

\arctan (x)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({\arctan (x)}^{2})

2 \cdot \frac{d}{d x} (\arctan (x)) \cdot \arctan (x)

2 \cdot \frac{d}{d x} (\arctan (x)) \cdot \arctan (x)

ⓘ The derivative of arctan(x) is 1/(1 + x^2):

\frac{d}{d x} (\arctan (x))

\frac{1}{x^{2} + 1}

2 \cdot \frac{1}{x^{2} + 1} \cdot \arctan (x)

ⓘ Multiplying a fraction by a term multiplies its numerator:

2 \cdot \frac{1}{x^{2} + 1} \cdot \arctan (x)

\frac{2 \cdot 1 \cdot \arctan (x)}{x^{2} + 1}

\frac{2 \cdot \arctan (x)}{x^{2} + 1}

▶Derivative:

\frac{2 \cdot \arctan (x)}{x^{2} + 1}