https://file.aiquickdraw.com/mathbot/file/1726684674338srz3snok.jpg

We are asked to find the derivative of the function \( f(x) = (-6x + 7)e^{5x - 7} \).

### Step-by-step solution:

This function is a product of two parts:
1. \( (-6x + 7) \)
2. \( e^{5x - 7} \)

To differentiate, we apply the **product rule** for derivatives. The product rule states:

\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]

Let:
- \( f_1(x) = (-6x + 7) \)
- \( f_2(x) = e^{5x - 7} \)

Now, we calculate the derivatives of both parts:

1. **Derivative of \( f_1(x) = (-6x + 7) \):**
   \[
   f_1'(x) = -6
   \]

2. **Derivative of \( f_2(x) = e^{5x - 7} \):**
   Use the chain rule here. The derivative of \( e^{u} \) is \( e^{u} \cdot u' \), where \( u = 5x - 7 \).
   \[
   f_2'(x) = e^{5x - 7} \cdot 5 = 5e^{5x - 7}
   \]

### Now, applying the product rule:
\[
f'(x) = f_1'(x) \cdot f_2(x) + f_1(x) \cdot f_2'(x)
\]
Substitute the values:
\[
f'(x) = (-6) \cdot e^{5x - 7} + (-6x + 7) \cdot 5e^{5x - 7}
\]

Simplifying:
\[
f'(x) = -6e^{5x - 7} + 5(-6x + 7)e^{5x - 7}
\]

Factor out \( e^{5x - 7} \):
\[
f'(x) = e^{5x - 7} \left( -6 + 5(-6x + 7) \right)
\]

Simplify the expression inside the parentheses:
\[
f'(x) = e^{5x - 7} \left( -6 - 30x + 35 \right)
\]
\[
f'(x) = e^{5x - 7} \left( -30x + 29 \right)
\]

Thus, the derivative is:
\[
f'(x) = (-30x + 29) e^{5x - 7}
\]

Would you like more details, or do you have any questions?

---

Here are five related questions to deepen your understanding:
1. How does the product rule apply to more complex functions?
2. What is the significance of the chain rule in differentiating exponential functions?
3. Can you differentiate \( f(x) = x^2 e^{3x} \) using a similar approach?
4. How does factoring out common terms simplify derivative expressions?
5. How would you interpret the derivative of this function graphically?

**Tip:** Always look for common terms in your derivative expression to simplify and factorize where possible.

Soit la fonction f(x) = (-6x + 7) * e^(5x - 7). Déterminer la dérivée de f.

Learn how to compute the derivative of the function f(x) = (-6x + 7) * e^(5x - 7) by applying the product rule and chain rule. A step-by-step solution is provided, ideal for undergraduate 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