# Derivative of $\tan{\left(x^{2} \right)}$

The calculator will find the derivative of $\tan{\left(x^{2} \right)}$, with steps shown.

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### Your Input

Find $\frac{d}{dx} \left(\tan{\left(x^{2} \right)}\right)$.

### Solution

The function $\tan{\left(x^{2} \right)}$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = \tan{\left(u \right)}$ and $g{\left(x \right)} = x^{2}$.

Apply the chain rule $\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$:

$${\color{red}\left(\frac{d}{dx} \left(\tan{\left(x^{2} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\tan{\left(u \right)}\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$

The derivative of the tangent is $\frac{d}{du} \left(\tan{\left(u \right)}\right) = \sec^{2}{\left(u \right)}$:

$${\color{red}\left(\frac{d}{du} \left(\tan{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(\sec^{2}{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right)$$

Return to the old variable:

$$\sec^{2}{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) = \sec^{2}{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right)$$

Apply the power rule $\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$ with $n = 2$:

$$\sec^{2}{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = \sec^{2}{\left(x^{2} \right)} {\color{red}\left(2 x\right)}$$

Thus, $\frac{d}{dx} \left(\tan{\left(x^{2} \right)}\right) = 2 x \sec^{2}{\left(x^{2} \right)}$.

### Answer

$\frac{d}{dx} \left(\tan{\left(x^{2} \right)}\right) = 2 x \sec^{2}{\left(x^{2} \right)}$A