Math Problem Statement

Use the figures below to evaluate the indicated derivative, or state that it does not exist. If the derivative does not exist, enter DNE. The graph to the left (in black) gives f(x), while the graph to the right gives g(x). Calculate the derivative d/dx[f(g(x))] at x = 100.

Solution

We are tasked with finding the value of the derivative ddxf(g(x))\frac{d}{dx} f(g(x)) at x=100x = 100, based on the graphs of f(x)f(x) and g(x)g(x) provided.

Here's how we can approach this problem:

Step 1: Chain Rule

The derivative ddxf(g(x))\frac{d}{dx} f(g(x)) is computed using the chain rule: ddxf(g(x))=f(g(x))g(x)\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) So, to evaluate this at x=100x = 100, we need two things:

  1. The value of g(100)g(100) to find f(g(100))f'(g(100)).
  2. The value of g(100)g'(100).

Step 2: Analyze the Graph of g(x)g(x)

From the graph of g(x)g(x) (the graph on the right in blue), we see that:

  • At x=100x = 100, the graph is a horizontal line, meaning g(x)g(x) is constant in this region.
  • Therefore, g(100)=0g'(100) = 0.

Step 3: Compute the Derivative

Since g(100)=0g'(100) = 0, regardless of the value of f(g(100))f'(g(100)), the overall derivative is: f(g(100))g(100)=f(g(100))0=0f'(g(100)) \cdot g'(100) = f'(g(100)) \cdot 0 = 0

Thus, the value of the derivative ddxf(g(x))\frac{d}{dx} f(g(x)) at x=100x = 100 is: 00

Final Answer:

ddxf(g(x))x=100=0\frac{d}{dx} f(g(x))\bigg|_{x=100} = 0

Let me know if you'd like further clarification or details!


Here are 5 related questions to expand on this:

  1. What happens to ddxf(g(x))\frac{d}{dx} f(g(x)) when g(x)g'(x) is non-zero?
  2. How does the chain rule apply when f(x)f(x) has a discontinuity?
  3. What is the derivative of a composition of three functions, f(h(g(x)))f(h(g(x)))?
  4. How can we find the derivative of g(x)g(x) when it's piecewise linear?
  5. How does the concept of derivative apply to motion in physics?

Tip: The chain rule is essential in dealing with composite functions, so practice identifying the inner and outer functions carefully!

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Math Problem Analysis

Mathematical Concepts

Calculus
Chain Rule
Composite Functions
Derivatives

Formulas

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

Theorems

Chain Rule

Suitable Grade Level

Grades 11-12