Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.2: Derivatives Of Products And Quotients
Problem 35E
Related questions
Question
![**Problem Description:**
Given the function \( f(x) = -5x(x - 1) \),
1. Compute the derivative \( f'(-3) \).
**Answer Box:**
2. Simplify and then find the general form of the derivative \( f'(x) \).
**Answer Box:**
**Hint:** You may want to expand and simplify the expression for \( f(x) \) first.
**Step-by-Step Solution:**
1. **Expand the Function \( f(x) \):**
To find \( f(x) \), first expand the given function:
\[
f(x) = -5x(x - 1) = -5x^2 + 5x
\]
2. **Differentiate \( f(x) \):**
Now differentiate \( f(x) \) with respect to \( x \):
\[
f'(x) = \frac{d}{dx}(-5x^2 + 5x) = -10x + 5
\]
3. **Substitute \( x = -3 \) into the derivative:**
To find \( f'(-3) \), substitute \( x = -3 \) into \( f'(x) \):
\[
f'(-3) = -10(-3) + 5 = 30 + 5 = 35
\]
Therefore:
- \( f'(-3) = 35 \)
- After simplifying, \( f'(x) \) is \( -10x + 5 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3972ee88-18a6-4abd-967b-f522c80949cc%2F733ae7ed-6e28-40c3-9396-92ab1bc065ec%2Fz8inp0i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Given the function \( f(x) = -5x(x - 1) \),
1. Compute the derivative \( f'(-3) \).
**Answer Box:**
2. Simplify and then find the general form of the derivative \( f'(x) \).
**Answer Box:**
**Hint:** You may want to expand and simplify the expression for \( f(x) \) first.
**Step-by-Step Solution:**
1. **Expand the Function \( f(x) \):**
To find \( f(x) \), first expand the given function:
\[
f(x) = -5x(x - 1) = -5x^2 + 5x
\]
2. **Differentiate \( f(x) \):**
Now differentiate \( f(x) \) with respect to \( x \):
\[
f'(x) = \frac{d}{dx}(-5x^2 + 5x) = -10x + 5
\]
3. **Substitute \( x = -3 \) into the derivative:**
To find \( f'(-3) \), substitute \( x = -3 \) into \( f'(x) \):
\[
f'(-3) = -10(-3) + 5 = 30 + 5 = 35
\]
Therefore:
- \( f'(-3) = 35 \)
- After simplifying, \( f'(x) \) is \( -10x + 5 \).
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