Question

Find the derivatives of the function:$f\left(x\right)=x^{3}secx.$

Short answer

The derivative of $f\left(x\right)=x^{3}sec\left(x\right)is{x}^2\{3sec\left(x\right)+x\tan \left(x\right)sec\left(x\right)\}.$

Step-by-step solution

Given information  

The given expression is$f\left(x\right)=x^{3}secx.$

Simplification    

$$\begin{gathered} f\text{'}\left(x\right) \\ =\frac{d}{dx}\left(x^{3}sec\left(x\right)\right) \\ =sec\left(x\right)\frac{d}{dx}\left(x^3\right)+x^3\frac{d}{dx}\left(sec\left(x\right)\right) \\ =3x^2\times \frac{1}{\cos \left(x\right)}+x^3\frac{\sin x}{{\cos }^2\left(x\right)} \\ =x^2\{3sec\left(x\right)+x\tan \left(x\right)sec\left(x\right)\}. \\ Hence,f\text{'}\left(x\right)=x^2\{3sec\left(x\right)+x\tan \left(x\right)sec\left(x\right)\}. \end{gathered}$$