Question

calculate the derivative of the following function. y = tan(e^x) dy/dx = □

Explanation:

Step1: Identify the outer and inner functions

Let $u = e^{x}$, then $y=\tan(u)$.

Step2: Find the derivative of the outer function

The derivative of $\tan(u)$ with respect to $u$ is $\sec^{2}(u)$. So, $\frac{dy}{du}=\sec^{2}(u)$.

Step3: Find the derivative of the inner function

The derivative of $u = e^{x}$ with respect to $x$ is $\frac{du}{dx}=e^{x}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=\sec^{2}(u)$ and $\frac{du}{dx}=e^{x}$, and replacing $u = e^{x}$, we get $\frac{dy}{dx}=e^{x}\sec^{2}(e^{x})$.

Answer:

$e^{x}\sec^{2}(e^{x})$