What is the Derivative of e^x^3?

Derivative of e^x^3

The derivative of

e^{x^3}

3e^{x^3} x^2

.

Solution. Let

F(x) = e^{x^3}

f(u) = e^{u}

and

g(x) = x^3

such that

F(x) = f(g(x))

. We will use the chain rule:

\begin{align*} F'(x) = f'(g(x))g'(x). \end{align*}

Since we have proven here that

f'(u) = e^{u}

and

g'(x) = 3x^2

, we get:

\begin{align*} f'(g(x)) = e^{g(x)} = e^{x^3} \quad \text{and} \quad g'(x) = 3x^2. \end{align*}

Therefore, we get:

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= e^{x^3} \cdot 3x^2 \\ &= 3e^{x^3} x^2. \end{align*}

So, the derivative of

e^{x^3}

3e^{x^3} x^2