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> c. p ′ ( 4 ) p^{\prime}\left(4\right)

Welcome back, everyone. Let KFX be equal to G H X. Calculate K at point2 using the following table, and we're given for answer choices A -7, B 0, C 14, and D-28. So first of all, let's understand that KFX is a composite function. So its derivative is going to be found using the chain rule. We have to take the derivative of the outer function G at H X and multiply it by the derivative of the inner function, the derivative of H X. And now if we're looking at K at 1.2, it is equal to G at H2 because that's our X multiplied by H at.2. So the first thing that we need to find is H at 0.2, and if we look at the table, H of X intersects with X equals 2 at 2. So H of 2 is equal to 2, right? Now our expression simplifies to K at 0.2 is G at 0.2. Multiplied by H at. 2. Now we need to find G at 0.2 and based on the table, G at. 2 is equal to -4. And we also need H at 0.2. Each prime at 0.2 is equal to 7. So now we can substitute these. K at 0.2 is equal to -4, multiplied by 7, which gives us -28. So the final answer to this problem corresponds to the answer choice D. Thank you for watching.

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

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Problem 3.7.25c

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>
c. $p prime of 4$

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Identify the function composition for p(x) = g(f(x)). We need to find the derivative p'(x) using the chain rule.

Apply the chain rule for derivatives: If p(x) = g(f(x)), then p'(x) = g'(f(x)) * f'(x).

Evaluate f(x) at x = 4 to find f(4). Use the table to find the value of f(4).

Use the table to find f'(4), the derivative of f at x = 4.

Substitute f(4) into g'(f(x)) to find g'(f(4)) using the table, then multiply g'(f(4)) by f'(4) to find p'(4).

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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'(4), represents the rate of change of a function at a specific point. The notation p'(4) indicates the derivative of the function p(x) evaluated at x = 4. Understanding this notation is crucial for interpreting the results of differentiation and applying them to specific values.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For example, evaluating p(4) means substituting 4 into the function p(x) to find its value. This concept is important when calculating derivatives, as it often requires evaluating the original functions at certain points to find the necessary derivatives for applying the Chain Rule.