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> d. p ′ ( 2 ) p^{\prime}\left(2\right)

Welcome back, everyone. Let K of X be equal to G H X. Calculate K at 0.1 using the following table, and we're given our table and for answer choices A -2, B -1, C0, and D2. So essentially for this problem we want to apply the chain rule, right, because K of X is a composite function. So if we want to find the general form of the derivative K of X, we're going to differentiate G H X, right? Because this is how K of X is defined. According to the chain rule, well, essentially, we are going to take the derivative of G with respect to H of X. And then multiply the result by the derivative of the inner function, so H of x. And now if we are looking at X equals 1, then we can say that the derivative of k at 0.1 is simply equal to. The derivative of g at age of 1 we're substituting 1 right multiplied by the derivative of H at 0.1. So now we understand what we want to find, right? First of all, we can see that we need. H.1. So if we look at the table, we can see H of X, and if we look at X equals 1, this gives us a value of 3, right? So let's label our first value that is going to be equal to 3. And this means we can simplify. This expression, we're going to get G. Prime At 0.3 because H at 0.1 is equal to 3, multiplied by H at 0.1. So now we only need to use two values. Let's look at G and let's identify its value at 0.3. According to the table, this is -1. And now each prime at 0.1. Is going to be equal to 2 according to the table, right, so -1. Is multiplied by 2. And this gives us our final result, which is -2, -1 multiplied by 2 is -2. So our final answer corresponds to the answer choice A. Thank you for watching.

2. Intro to Derivatives

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

2:44 minutes

Problem 3.7.25d

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.
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d. $p prime of 2$

Verified step by step guidance

Identify that you need to find the derivative of the composite function p(x) = g(f(x)) at x = 2, which is p'(2).

Recall the chain rule for derivatives, which states that if you have a composite function p(x) = g(f(x)), then the derivative p'(x) = g'(f(x)) * f'(x).

Evaluate f(x) at x = 2 using the table to find f(2). This will give you the input for g'.

Use the table to find g'(f(2)), which is the derivative of g at the point f(2).

Find f'(2) using the table, which is the derivative of f at x = 2. Multiply g'(f(2)) by f'(2) to get p'(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 and g, such that h(x) = f(g(x)), then the derivative h'(x) can be found using the formula h'(x) = f'(g(x)) * g'(x). This rule is essential for calculating derivatives of functions that are nested within each other.

Derivative Notation

Derivative notation, such as f'(x) or p'(2), represents the rate of change of a function with respect to its variable. The notation p'(2) specifically indicates the derivative of the function p evaluated at the point x = 2. Understanding this notation is crucial for interpreting and calculating derivatives accurately.

Function Composition

Function composition occurs when one function is applied to the result of another function. In the context of the question, h(x) = f(g(x)) and p(x) = g(f(x)) are examples of composed functions. Recognizing how to work with composed functions is vital for applying the Chain Rule and finding derivatives of such functions.

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