Derivatives using tables Let h ( x ) = f ( g ( x ) ) h(x)=f(g(x)) h ( x ) = f ( g ( x )) and p ( x ) = g ( f ( x ) ) p(x)=g(f(x)) p ( x ) = g ( f ( x )) . Use the table to compute the following derivatives. <IMAGE> b. h ′ ( 2 ) h^{\prime}\left(2\right)

Welcome back, everyone. Let F of X be equal to H of G of X. Calculate F at 0.0 using the following table. We're given 4 answer choices A -24, B-21, C7, and D 8. So first of all, we're given a composite function. Let's find its derivative. F of X is going to be equal to H G X multiplied by G of X based on the chain rule, right? So we're taking the outer function, it's derivative multiplied by the derivative of the inner function. And now, if we're looking at 0.0, we are simply getting F at 0 equals H G 0 multiplied by G at 0.0. So now what we're going to do is identify G at 0.0. So we're looking at the table, we identify G of X at 0.0. It is equal to 2. So we can now simplify the expression. F at 0.0 is H at 0.2 multiplied by G at 0.0. Now let's identify H at 0.2. Each prime at 0.2 based on the table is equal to 7. And we also need G at 0.0. So now G at 0.0 is -3. And then we have everything that we need, so F at 0.0 is simply 7 multiplied by -3. Which is -21. And this means that the final answer to this problem corresponds to the answer choice B. Thank you for watching.

2. Intro to Derivatives

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

2:08 minutes

Problem 3.7.25b

Textbook Question

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$ . Use the table to compute the following derivatives.
<IMAGE>
b. $h prime of 2$

Verified step by step guidance

Identify that h(x) = f(g(x)) is a composition of functions, which requires the use of the chain rule to find its derivative.

Recall the chain rule: if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).

To find h'(2), substitute x = 2 into the derivative expression: h'(2) = f'(g(2)) * g'(2).

Use the table to find the values of g(2) and g'(2). Substitute these values into the expression.

Next, use the table to find f'(g(2)) by first finding g(2) and then using this result to find f' at that point. Substitute this value into the expression to complete the calculation of h'(2).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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'(2) specifically refers to the derivative of the function h evaluated at x = 2. 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 certain points, such as h(2) or g(f(2)), 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.

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