Question

Using Relationships In Exercises \(103-106,\) use the given information to find \(f^{\prime}(2) .\) $$ \begin{array}{l}{g(2)=3 \quad \text { and } \quad g^{\prime}(2)=-2} \\\ {h(2)=-1 \quad \text { and } \quad h^{\prime}(2)=4}\end{array} $$ $$ f(x)=4-h(x) $$

Short answer

Hence, \(f'(2) = -4\).

Step-by-step solution

Determine Function \(h'(x)\)

Firstly, it's necessary to define \(h'(x)\), which is the derivative of \(h(x)\). We know that the value of \(h'(2)\) equals 4.

Apply Chain Rule to \(f(x)\)

We need to find \(f'(x)\), the derivative of \(f(x)\). To do so, let's use the chain rule, which states that the derivative of a compound function is the derivative of the outside function times the derivative of the inside function. Apply chain rule to function \(f(x) = 4 - h(x)\). Its derivative is \(f'(x) = -h'(x)\). We replace \(x\) with 2 to find \(f'(2)\).

Calculate \(f'(2)\)

As we know from step 1 that \(h'(2) = 4\), substitute this value into equation from step 2. This calculation yields \(f'(2) = -h'(2) = -4\).

Key concepts

Understanding Function Derivative

Derivatives in calculus play a key role when we talk about how a function changes. A derivative is essentially the rate at which one quantity changes with respect to another. For simple functions like polynomials, finding the derivative is straightforward with basic differentiation rules. However, for more complex functions, especially those involving other functions, it becomes trickier.When we apply differentiation to a function like \(f(x)\), we look for its derivative \(f'(x)\). This derivative tells us about the behavior of the function at any given point. If a function is given by an expression involving other functions, like \(f(x) = 4 - h(x)\), finding the derivative involves additional steps because of the nested nature.

Basics of Derivative Calculation

In calculus, calculating the derivative means finding out how a function behaves as its input values change. To perform derivative calculations, we typically use rules like the power rule, product rule, quotient rule, and chain rule. The chain rule is particularly important when dealing with functions built out of other functions, known as compound functions.The problem at hand involves a simple yet important use of the derivative calculation. We need to determine \(f'(2)\) given that \(f(x) = 4 - h(x)\). By understanding that \(h(x)\) is another function, we can apply differentiation rules effectively to deduce \(f'(x)\). Substituting any known values, such as \(h'(2)\), helps simplify the outcome of our calculation.

Compound Function Differentiation With Chain Rule

The chain rule is an essential tool in calculus for differentiating compositions of functions. This rule states that to differentiate a compound function, you must take the derivative of the outer function and multiply it by the derivative of the inner function.Here, we use the chain rule on \(f(x) = 4 - h(x)\). Although \(4\) is a constant and does not change, \(h(x)\) involves another variable. The derivative here, according to the chain rule, becomes \(f'(x) = -h'(x)\). This underscores the fact that the negative sign comes from the subtraction operation applied to \(h(x)\).Once the rule is applied, substituting \(h'(2) = 4\) gives us the result \(f'(2) = -4\). This process not only finds the derivative but also shows the dependency of \(f'(x)\) on another function's behavior, highlighting the power of the chain rule in managing complexity.