differentiate the following function.\ny = e^{x^{3}}\ny = \\frac{d}{dx}(e^{x^{3}})=\\square

Explanation:

Step1: Apply chain - rule

Let $u = x^{3}$, then $y = e^{u}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Differentiate $y$ with respect to $u$

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

Step3: Differentiate $u$ with respect to $x$

Since $u = x^{3}$, then $\frac{du}{dx}=3x^{2}$.

Step4: Calculate $\frac{dy}{dx}$

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=e^{u}\cdot3x^{2}$. Substituting $u = x^{3}$ back in, we get $\frac{dy}{dx}=3x^{2}e^{x^{3}}$.

Answer:

$3x^{2}e^{x^{3}}$