Problem 245 - Calculus Volume 1

Question

Use the information in the following table to find \(h^{\prime}(a)\) at the given value for \(a\). $$\begin{array}{|c|c|c|c|c|} \hline x & f(x) & f^{\prime}(x) & g(x) & g^{\prime}(x) \\ \hline 0 & 2 & 5 & 0 & 2 \\ \hline 1 & 1 & -2 & 3 & 0 \\ \hline 2 & 4 & 4 & 1 & -1 \\ \hline 3 & 3 & -3 & 2 & 3 \\ \hline \end{array}$$ $$ h(x)=f(g(x)) ; a=0 $$

Step-by-Step Solution

Answer

The derivative \( h^{\prime}(0) \) is 10.

Step 1: Understand the Function

The function we need to differentiate is a composite function, \( h(x) = f(g(x)) \). This means that \( h(x) \) is a function \( f \) applied to \( g(x) \). In this problem, we need to find \( h^{\prime}(a) \) for \( a = 0 \).

Step 2: Apply the Chain Rule for Derivatives

To find the derivative of a composite function \( h(x) = f(g(x)) \), we use the chain rule which states: \( h^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x) \). We will use this formula to find \( h^{\prime}(0) \).

Step 3: Evaluate Inside Function at \(a = 0\)

First, we need to find \( g(0) \), as \( g(x) \) is inside the function \( f(g(x)) \). From the table, when \( x = 0 \), \( g(0) = 0 \).

Step 4: Differentiate Outer Function at \(g(a)\)

We now need \( f^{\prime}(g(0)) \). Since \( g(0) = 0 \), we find \( f^{\prime}(0) \). From the table, \( f^{\prime}(0) = 5 \).

Step 5: Derivative of Inside Function

The next step is to find \( g^{\prime}(0) \) from the table, which is given as \( g^{\prime}(0) = 2 \).

Step 6: Calculate \(h^{\prime}(0)\)

Combine these results using the chain rule: \( h^{\prime}(0) = f^{\prime}(g(0)) \cdot g^{\prime}(0) = 5 \cdot 2 = 10 \).

Step 7: Conclusion

Therefore, the derivative \( h^{\prime}(0) \) is \( 10 \).

Key Concepts

Composite Functions

Composite functions play a vital role in calculus, especially when dealing with derivatives. Imagine a situation where you have two functions, say \( f(x) \) and \( g(x) \). A composite function is where one function is applied to the result of another function. This is expressed as \( h(x) = f(g(x)) \). In this scenario, you plug the output of \( g(x) \) into the function \( f(x) \).

Composite functions are like mathematical nesting dolls where one function fits inside another. Understanding the structure of composite functions is essential for solving problems related to derivatives, such as finding the rate of change in complicated systems or processes.

In the exercise, the composite function is \( h(x) = f(g(x)) \). Essentially, you are observing how \( h(x) \) behaves when \( g(x) \) is input through \( f(x) \). This concept forms the groundwork for applying the chain rule, which helps in differentiating such functions.

Chain Rule

The chain rule is an invaluable tool in calculus for differentiating composite functions. It allows you to find the derivative, \( h^{\prime}(x) \), of a composite function \( h(x) = f(g(x)) \). The chain rule formula is:

• \( h^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x) \)

Essentially, this rule is about breaking the derivative into two parts: the derivative of the outer function at the inner function, \( f^{\prime}(g(x)) \), and the derivative of the inner function, \( g^{\prime}(x) \).

To apply the chain rule effectively, first evaluate the inside function \( g(x) \), and then find the derivative of the outer function, \( f(x) \), at this evaluated point. The next step is to find the derivative of the inside function itself.

In our exercise, we utilized the chain rule to differentiate \( h(x) = f(g(x)) \) at \( a = 0 \). We considered \( f^{\prime}(g(0)) \) and \( g^{\prime}(0) \) from the table to compute \( h^{\prime}(0) \), ensuring that the calculations were in line with the chain rule's strategy.

Function Differentiation

Differentiating a function is one of the fundamental practices in calculus. It involves finding the rate at which a function value changes as its input changes. Differentiation is like finding the speed of a car on a highway; instead of miles per hour, you are determining the rate of change per unit.

In a typical differentiation problem, you determine \( f^{\prime}(x) \), the derivative of \( f(x) \), using standard rules of calculus, like the power rule, product rule, or chain rule. Each of these has its unique way of handling different mathematical expressions.

For composite functions like \( h(x) = f(g(x)) \), differentiation isn't straightforward, requiring the use of the chain rule. This method ensures we correctly account for how changes in \( g(x) \) affect \( f(x) \). By applying these principles correctly, we calculated \( h^{\prime}(0) = 10 \), confirming the precise approach to differentiation of composite functions.