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

Let's differentiate:

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

We differentiate using:

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

(where f =

\arctan (2 x)

)

ⓘ The derivative of x^n is n*x^(n-1):

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

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

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

Apply the differentiation rule:

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

with f =

2 x

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

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

\frac{\frac{d}{d x} (2 x)}{{(2 x)}^{2} + 1}

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

■A constant factor stays in front of the derivative, (k*u)' = k*u':

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

2

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

ⅰ

2 \cdot 2

4

\frac{4 \cdot \arctan (2 x)}{{(2 x)}^{2} + 1}

ⓘ The power of a product is the product of the powers:

{(2 x)}^{2}

2^{2} \cdot x^{2}

\frac{4 \cdot \arctan (2 x)}{2^{2} \cdot x^{2} + 1}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{2}

2 \cdot 2

4

▶Derivative:

\frac{4 \cdot \arctan (2 x)}{4 x^{2} + 1}