Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.
<IMAGE>
e. $h^{\prime }\left(5\right)$

Verified step by step guidance

1. Identify the function composition: h(x) = f(g(x)). This is a composition of functions where f is the outer function and g is the inner function.
2. Recall the chain rule for derivatives, which states that if you have a composition of functions h(x) = f(g(x)), then the derivative h'(x) is given by h'(x) = f'(g(x)) * g'(x).
3. To find h'(5), substitute x = 5 into the chain rule formula: h'(5) = f'(g(5)) * g'(5).
4. Use the table to find the values of g(5) and g'(5). Substitute these values into the expression for h'(5).
5. Next, use the table to find the value of f' at the point g(5). Substitute this value into the expression for h'(5) to complete the calculation.

Key Concepts

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function h(x) is composed of two functions, f(g(x)), the derivative h'(x) can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. Mathematically, this is expressed as h'(x) = f'(g(x)) * g'(x). Understanding this rule is essential for solving problems involving derivatives of composite functions.

Derivative Notation

Derivative notation, such as h'(x) or f'(x), represents the rate of change of a function with respect to its variable. It indicates how the function's output changes as its input changes. In the context of the question, h'(5) specifically refers to the derivative of the function h evaluated at x = 5. Familiarity with this notation is crucial for interpreting and calculating derivatives correctly.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of derivatives, evaluating functions at particular points, such as h(5) or g(f(5)), is necessary to compute the derivative using the Chain Rule. This concept is vital for applying the derivatives obtained from the Chain Rule to find specific values, which is often required in calculus problems.