Problem 72 - Calculus an Applied Approach

Question

Use the given information to find \(f^{\prime}(2) .\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=\frac{g(x)}{h(x)} $$

Step-by-Step Solution

Answer

The value of \(f'(2)\) is -10.

Step 1: Identify the Derivative Formula for a Ratio

The derivative of a function given as a ratio \(f(x)=g(x)/h(x)\) can be found using the formula \(f'(x)= (g'(x)h(x) - g(x)h'(x))/(h(x))^2\).

Step 2: Apply Given Values

Substitute the given values into the derivative formula: \(f'(2)= (g'(2)h(2) - g(2)h'(2))/(h(2))^2\), where \(g(2)=3\), \(g'(2)=-2\), \(h(2)=-1\), and \(h'(2)=4\).

Step 3: Complete the Calculation

Perform the calculations to find \(f'(2) = ((-2*(-1)) - (3*4))/((-1)^2) = (2 - 12)/1 = -10\).

Key Concepts

Differentiation

In calculus, differentiation is a major concept that refers to the process of finding the derivative of a function. The derivative represents how a function's output value changes in response to changes in its input value. Essentially, a derivative provides an instant rate of change of the function at a given point. It's like taking a snapshot of the function's movement (rate of change) at an exact instant in time.

Differentiation can be applied to various kinds of functions, and there are specific rules and formulas to handle different scenarios. For example, when differentiating polynomials, power rules are used, while for products of functions, the product rule is applied. But what about when a function is defined as a ratio of two other functions? In such cases, the quotient rule comes into play, which is a specific technique for differentiating ratios of functions.

Quotient Rule

When it comes to differentiation, the quotient rule is specifically designed for finding the derivative of a function that is expressed as the ratio of two functions. The rule states that if you have a function in the form of \(f(x) = \frac{g(x)}{h(x)}\), where both \(g(x)\) and \(h(x)\) are differentiable functions, the derivative of \(f(x)\) is given by:

\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \]

This formula requires you to take the derivatives of the numerator and the denominator functions separately (\(g'(x)\) and \(h'((x)\)), multiply them with the opposite function (respectively \(h\) and \(g\)), and subtract one product from the other. Finally, you square the denominator function and place the result beneath the subtraction you performed.

Chain Rule

The chain rule is another fundamental technique in differentiation that comes into play when dealing with composite functions - functions made up of two or more functions. For instance, if \(f(x)\) is the composite of \(g(x)\) and \(h(x)\), where \(f(x) = g(h(x))\), then the chain rule helps us find the derivative of \(f\).

The chain rule states:

\[ f'(x) = g'(h(x)) \cdot h'(x) \]

This means, to find the derivative of \(f\), you first need to differentiate the outer function \(g\) with respect to \(h\), and then multiply it by the derivative of the inner function \(h\) with respect to \(x\). The chain rule is invaluable when functions are not simply sums, products, or quotients, but are instead functions within other functions, creating a 'chain' of functions.