To derive the function $f(x) = \sqrt{e^{x^3} + x}$, we will apply the chain rule and other differentiation techniques. Here's the step-by-step process:

1. Express the function

Given: $f(x) = \sqrt{e^{x^3} + x}$ This can also be written as: $f(x) = (e^{x^3} + x)^{1/2}$

2. Differentiate using the chain rule

To differentiate this, we'll apply the chain rule. The chain rule states: $\frac{d}{dx} \left( g(h(x)) \right) = g'(h(x)) \cdot h'(x)$ In this case:

• $g(x) = x^{1/2}$ (the outer function)
• $h(x) = e^{x^3} + x$ (the inner function)

First, differentiate the outer function $g(h(x)) = (h(x))^{1/2}$: $\frac{d}{dx} \left( h(x)^{1/2} \right) = \frac{1}{2} \cdot (h(x))^{-1/2}$

Now apply the chain rule: $f'(x) = \frac{1}{2} \cdot (e^{x^3} + x)^{-1/2} \cdot \frac{d}{dx} \left( e^{x^3} + x \right)$

3. Differentiate the inner function

Next, differentiate the inner function $h(x) = e^{x^3} + x$.

• For $e^{x^3}$, apply the chain rule again. The derivative of $e^{x^3}$ is: $\frac{d}{dx} \left( e^{x^3} \right) = e^{x^3} \cdot \frac{d}{dx} \left( x^3 \right) = e^{x^3} \cdot 3x^2$

• The derivative of $x$ is simply 1.

Thus, the derivative of $h(x) = e^{x^3} + x$ is: $h'(x) = 3x^2 e^{x^3} + 1$

4. Combine the results

Now plug this back into the chain rule expression for $f'(x)$: $f'(x) = \frac{1}{2} \cdot (e^{x^3} + x)^{-1/2} \cdot (3x^2 e^{x^3} + 1)$

Final derivative:

$f'(x) = \frac{3x^2 e^{x^3} + 1}{2 \sqrt{e^{x^3} + x}}$

This is the derivative of $f(x) = \sqrt{e^{x^3} + x}$.

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Would you like a further breakdown or additional details?

Here are 5 related questions to explore:

1. How does the chain rule apply to multi-variable functions?
2. What is the general rule for differentiating exponential functions like $e^{g(x)}$?
3. How do you differentiate functions of the form $\sqrt{u(x)}$?
4. What are the common errors when applying the chain rule?
5. Can you apply the product rule together with the chain rule in more complex cases?

Tip: When differentiating composite functions, it's helpful to break the function into outer and inner components and apply the chain rule systematically.