How to derive f(x)= square root of e^x^3+x

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 \( f(x) = \sqrt{e^{x^3} + x} \).

---

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.

How to derive f(x)= square root of e^x^3 + x

Learn how to differentiate the function f(x) = sqrt(e^x^3 + x) using the chain rule and exponential differentiation techniques. This problem involves applying the chain rule, handling composite functions, and differentiating exponential expressions. Suitable for students in undergraduate calculus or advanced high school calculus.

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