https://file.aiquickdraw.com/mathbot/file/1726406478163t73q42w8.jpg

To find the derivative of the given function:

\[
g(x) = \frac{5x^3}{e^x}
\]

we can use the **quotient rule**. The quotient rule states that if \( g(x) = \frac{f(x)}{h(x)} \), then:

\[
g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{[h(x)]^2}
\]

For this problem:
- \( f(x) = 5x^3 \) and \( h(x) = e^x \).
  
### Step-by-Step Solution

1. **Differentiate \( f(x) = 5x^3 \):**

\[
f'(x) = 15x^2
\]

2. **Differentiate \( h(x) = e^x \):**

\[
h'(x) = e^x
\]

3. **Apply the quotient rule:**

\[
g'(x) = \frac{(15x^2)(e^x) - (5x^3)(e^x)}{(e^x)^2}
\]

4. **Simplify the numerator:**

\[
g'(x) = \frac{e^x(15x^2 - 5x^3)}{e^{2x}}
\]

5. **Further simplification:**

Since there is an \( e^x \) in both the numerator and the denominator, we can cancel one \( e^x \) term:

\[
g'(x) = \frac{15x^2 - 5x^3}{e^x}
\]

6. **Factor out a common term in the numerator:**

\[
g'(x) = \frac{5x^2(3 - x)}{e^x}
\]

Thus, the derivative of the function is:

\[
g'(x) = \frac{5x^2(3 - x)}{e^x}
\]

Would you like further clarification or have any questions?

Here are 5 related questions to consider:
1. How does the quotient rule differ from the product rule in differentiation?
2. What is the significance of the exponential function \( e^x \) in calculus?
3. How would the derivative change if the numerator were a different polynomial?
4. Can the expression \( \frac{5x^2(3 - x)}{e^x} \) be simplified further?
5. How would you find the critical points of the function \( g(x) \) using the derivative?

**Tip:** Always look for common factors in the numerator when simplifying derivatives; it can make the expression more manageable!

Find the derivative of the function g(x) = 5x^3/e^x.

Learn how to find the derivative of the function g(x) = 5x^3/e^x using the quotient rule. This calculus problem involves differentiating polynomial and exponential functions. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation