Q.Differentiate the following w.r.t. $x$: $e^{x^3}$

The key idea is the Chain Rule: differentiate the outer function (exponential) first, then multiply by the derivative of the inner function.

Step 1: Identify the outer function as $e^u$ and the inner function as $u = x^3$.

Step 2: Derivative of outer w.r.t. $u$: $\frac{d}{du} e^u = e^u$.

Step 3: Derivative of inner w.r.t. $x$: $\frac{du}{dx} = 3x^2$.

Step 4: Multiply: $\frac{d}{dx} e^{x^3} = e^{x^3} \cdot 3x^2$.

The key idea here is the Chain Rule. When you have a function of a function — like $e$ raised to something that itself depends on $x$ — you can't just differentiate the outer part and stop. You have to peel the layers like an onion: differentiate the outer layer, then multiply by the derivative of the inner layer.

Think of it this way: $e^{x^3}$ is the composition of two functions. The outer function is $f(u) = e^u$, and the inner function is $u(x) = x^3$. The Chain Rule says:

So we differentiate the outside (keeping the inside untouched), then multiply by the derivative of the inside.

Let's work through it step by step.

1. Identify the inner function.

   Here, the exponent $x^3$ is the "inside" part. Let $u = x^3$. Then our function becomes $e^u$.

2. Differentiate the outer function with respect to its argument.

   The derivative of $e^u$ with respect to $u$ is simply $e^u$ itself. So:

This means the derivative of the outer part, evaluated at $u = x^3$, is $e^{x^3}$.

3. Differentiate the inner function with respect to $x$. The derivative of $u = x^3$ is:

4. Multiply the two derivatives (Chain Rule). The Chain Rule tells us:

Substituting what we have:

5. Write the final result in standard form. It's conventional to write the constant factor first:

Method: The Chain Rule for a Composite Function

Whenever a function is "wrapped inside" another function — $f(g(x))$ — differentiate the outer function first (with respect to its own argument), then multiply by the derivative of the inner function.

Steps

Step 1: Identify the outer function and the inner function

Write $y=f(u)$ where $u=g(x)$ is everything "inside" the outermost operation.

Step 2: Differentiate the outer function with respect to $u$

Use the standard derivative rule for whatever the outer function is (log, power, trig, exponential, ...), keeping $u$ untouched.

Step 3: Differentiate the inner function $u$ with respect to $x$

Step 4: Multiply the two results

If the inner function is itself composite (a function inside a function inside a function), repeat the process — multiply in one more derivative for each layer.

Applying to this problem: for $y=e^{x^3}$, the outer function is $e^u$ with $u=x^3$; $\frac{dy}{du}=e^u=e^{x^3}$ (the exponential reproduces itself) and $\frac{du}{dx}=3x^2$, so $\frac{dy}{dx}=3x^2e^{x^3}$.

Common Mistakes

Mistake 1: Forgetting to multiply by the inner derivative $3x^2$, writing just $e^{x^3}$.

Why it's wrong: the derivative of $e^u$ with respect to $u$ is $e^u$ itself, but that is only the first factor of the chain rule — the inner function's own derivative must still be multiplied in. Correct approach: always ask "what is the derivative of what's in the exponent?" as a separate step.

Mistake 2: Applying the power rule ($x^n\to nx^{n-1}$) to $e^{x^3}$ instead of the exponential rule.

Why it's wrong: the power rule applies when the base is the variable and the exponent is constant — here it's the reverse (constant base $e$, variable exponent $x^3$), so the exponential chain rule is required instead. Correct approach: check which part of the expression is the variable before choosing power rule vs. exponential rule.