Math Problem Statement

If f'(-3) = -10, and g(x) = -8f(x), what is g'(-3)?

Solution

We are given the following information:

  • f(3)=10f'(-3) = -10
  • g(x)=8f(x)g(x) = -8f(x)

We need to find g(3)g'(-3).

Step 1: Differentiate g(x)g(x)

Given g(x)=8f(x)g(x) = -8f(x), we can differentiate both sides of this equation with respect to xx.

g(x)=8f(x)g'(x) = -8f'(x)

Step 2: Plug in the value x=3x = -3

Substitute f(3)=10f'(-3) = -10 into the derivative we just found:

g(3)=8(10)g'(-3) = -8(-10)

g(3)=80g'(-3) = 80

Final Answer:

g(3)=80g'(-3) = 80

Would you like more details or have any further questions?

Here are some related questions to consider:

  1. How do we differentiate functions using the chain rule?
  2. What does it mean to differentiate a function like g(x)=8f(x)g(x) = -8f(x)?
  3. How do we apply the product rule in cases with more complicated products?
  4. What happens when functions have constants multiplied by them?
  5. Can we use this method for other composite functions?

Tip: When differentiating a function multiplied by a constant, the constant remains and you only differentiate the variable function.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Function Derivatives
Calculus

Formulas

g'(x) = -8f'(x)

Theorems

Derivative of a function multiplied by a constant

Suitable Grade Level

Grades 11-12