Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

Find the derivatives with respect to x of the following combinations at the given value of x.

g. f(x + g(x)), x = 0

검증된 단계별 안내

Step 1: Recognize that the problem asks for the derivative of the composite function f(x + g(x)) at x = 0. To solve this, we will use the chain rule for derivatives.

Step 2: Recall the chain rule: If h(x) = f(u(x)), then h'(x) = f'(u(x)) * u'(x). Here, u(x) = x + g(x), so we need to compute the derivative of u(x) first.

Step 3: Compute u'(x). Since u(x) = x + g(x), its derivative is u'(x) = 1 + g'(x). At x = 0, g'(0) = 1/3, so u'(0) = 1 + 1/3 = 4/3.

Step 4: Apply the chain rule. The derivative of f(x + g(x)) at x = 0 is f'(u(0)) * u'(0). From the table, f'(x) at x = 0 is 5, and u(0) = 0 + g(0) = 0 + 1 = 1.

Step 5: Substitute the values into the chain rule expression. f'(u(0)) = f'(1), and from the table, f'(1) = -1/3. Combine this with u'(0) = 4/3 to complete the derivative calculation.

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Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. If you have a function h(x) = f(g(x)), the derivative h'(x) is found by multiplying the derivative of the outer function f at g(x) by the derivative of the inner function g at x, expressed as h'(x) = f'(g(x)) * g'(x). This rule is essential for solving problems involving nested functions.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. In the context of calculus, understanding how to compose functions is crucial for applying the chain rule. For example, if f(x) and g(x) are functions, the composition f(g(x)) means applying g first and then f to the result, which is key in differentiating composite functions.

Derivative Evaluation

Derivative evaluation involves calculating the derivative of a function at a specific point. This requires substituting the given x-value into the derivative function. In the problem, you need to evaluate the derivative of f(x + g(x)) at x = 0, using the values provided in the table for f, g, and their derivatives at x = 0, ensuring accurate computation of the derivative at that point.

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.Find the derivatives with respect to x of the following combinations at the given value of x.g. f(x + g(x)), x = 0