Question

question 19
find the derivative of $y = \arctan(5x^2 - 1)$.
\\(\boldsymbol{\frac{dy}{dx} = \frac{1}{25x^4 - 10x^2 + 2}}\\)
\\(\boldsymbol{\frac{dy}{dx} = \frac{1}{\sqrt{25x^4 - 10x^2 + 2}}}\\)
\\(\boldsymbol{\frac{dy}{dx} = \frac{10x}{25x^4 - 10x^2 + 2}}\\)
\\(\boldsymbol{\frac{dy}{dx} = \frac{10x}{\sqrt{25x^4 - 10x^2 + 2}}}\\)

Explanation:

Step1: Recall arctangent derivative rule

If $y=\arctan(u)$, then $\frac{dy}{dx}=\frac{1}{1+u^2}\cdot\frac{du}{dx}$

Step2: Define inner function $u$

Let $u=5x^2-1$, compute $\frac{du}{dx}$:
$\frac{du}{dx}=10x$

Step3: Compute $1+u^2$

$$\begin{align*} 1+u^2&=1+(5x^2-1)^2\\ &=1+25x^4-10x^2+1\\ &=25x^4-10x^2+2 \end{align*}$$

Step4: Substitute into chain rule

$\frac{dy}{dx}=\frac{1}{25x^4-10x^2+2}\cdot10x=\frac{10x}{25x^4-10x^2+2}$

Answer:

$\boldsymbol{\frac{dy}{dx} = \frac{10x}{25x^4 - 10x^2 + 2}}$ (the third option)