Use the given graphs of f and g to find each derivative. <IMAGE>
d/dx (g(f(x))) |x=1

Verified step by step guidance

1. Step 1: Recognize that you need to find the derivative of a composite function, g(f(x)), at x = 1. This requires the use of the chain rule.
2. Step 2: The chain rule states that the derivative of g(f(x)) with respect to x is g'(f(x)) * f'(x).
3. Step 3: Evaluate f(x) at x = 1 using the graph of f to find f(1).
4. Step 4: Use the graph of g to find g'(f(1)), which is the derivative of g at the point f(1).
5. Step 5: Use the graph of f to find f'(1), which is the derivative of f at x = 1. Multiply g'(f(1)) by f'(1) to find the derivative of g(f(x)) at x = 1.

Key Concepts

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if you have a function g(f(x)), the derivative is found by multiplying the derivative of the outer function g with the derivative of the inner function f. This rule is essential for solving problems involving nested functions, as it allows for the systematic breakdown of the differentiation process.

Derivative

A derivative represents the rate at which a function changes at a given point and is a core concept in calculus. It is defined as the limit of the average rate of change of the function as the interval approaches zero. Understanding derivatives is crucial for analyzing the behavior of functions, including their slopes, maxima, minima, and points of inflection.

Evaluating Derivatives at a Point

Evaluating derivatives at a specific point involves substituting the value of the variable into the derivative function. This process provides the instantaneous rate of change of the function at that point. In the context of the given question, finding d/dx (g(f(x))) at x=1 requires first applying the Chain Rule and then substituting x=1 into the resulting expression to obtain the final value.