Question

Derivative of $f\left(x\right)=\frac{e^x}{\cos \left(x^2\right)}$ is

• A. $\frac{e^x(\cos x^2+2x\sin x^2)}{\cos \left(x^2\right)}$
• B. $\frac{e^x(\cos x^2-2x\sin x^2)}{{\cos }^2\left(x^2\right)}$
• C. $\frac{e^x(\cos x^2+2x\sin x^2)}{{\cos }^2\left(x^2\right)}$
• D. $\frac{e^x(\cos x^2-2x\sin x^2)}{\cos \left(x^2\right)}$

Answer 1

The correct option is C $\frac{e^x(\cos x^2+2x\sin x^2)}{{\cos }^2\left(x^2\right)}$

Given, $f\left(x\right)=\frac{e^x}{\cos \left(x^2\right)}$
So, derivative of the given function is
$\frac{df\left(x\right)}{dx}=\frac{d\left(\frac{e^x}{\cos x^2}\right)}{dx}$
Using division and chain rule together we get
$\frac{\cos x^2\frac{d{e}^x}{dx}-e^x\frac{d\left(\cos x^2\right)}{d{x}^2}\times \frac{d{x}^2}{dx}}{(\cos x^2{)}^2}$
$\Rightarrow \frac{e^x.\cos x^2+2x{e}^x\sin x^2}{{\cos }^2\left(x^2\right)}$
$\Rightarrow \frac{e^x(\cos x^2+2x\sin x^2)}{{\cos }^2\left(x^2\right)}$