h(x)=(x²+3)³ determine the first derivative 

To find the first derivative of the function \( h(x) = (x^2 + 3)^3 \), we can apply the **chain rule**.

### Steps:

1. **Let** \( u(x) = x^2 + 3 \). Now, \( h(x) = u^3 \).
2. Use the chain rule:
   \[
   \frac{dh}{dx} = \frac{dh}{du} \cdot \frac{du}{dx}
   \]
   
   - First, differentiate \( u^3 \) with respect to \( u \): 
     \[
     \frac{dh}{du} = 3u^2
     \]
   
   - Next, differentiate \( u = x^2 + 3 \) with respect to \( x \):
     \[
     \frac{du}{dx} = 2x
     \]
   
3. Combine the two derivatives:
   \[
   \frac{dh}{dx} = 3u^2 \cdot 2x
   \]
   
4. Substitute \( u = x^2 + 3 \) back into the equation:
   \[
   \frac{dh}{dx} = 3(x^2 + 3)^2 \cdot 2x
   \]
   
5. Simplify the result:
   \[
   \frac{dh}{dx} = 6x(x^2 + 3)^2
   \]

Thus, the first derivative is:
\[
h'(x) = 6x(x^2 + 3)^2
\]

Would you like further details or clarification on any step?

### Related Questions:
1. How do you apply the chain rule to more complex functions?
2. What is the significance of the chain rule in calculus?
3. Can you explain how to find higher-order derivatives of \( h(x) = (x^2 + 3)^3 \)?
4. How would you find the second derivative of this function?
5. How does the power rule apply when differentiating polynomials?

### Tip:
When using the chain rule, identify the inner and outer functions first, and differentiate each part step by step.

h(x)=(x²+3)³ determine the first derivative

Learn how to find the first derivative of h(x) = (x² + 3)³ using the chain rule. This calculus problem explores key differentiation concepts, including the power rule and the chain rule, making it ideal for high school and college students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation