Problem 125 - Calculus Volume 1

Question

Assume that \(f(x)\) and \(g(x)\) are both differentiable functions for all \(x\). Find the derivative of each of the functions \(h(x)\). $$ h(x)=\frac{3 f(x)}{g(x)+2} $$

Step-by-Step Solution

Answer

The derivative is \( h'(x) = \frac{3(g(x) + 2)f'(x) - 3f(x)g'(x)}{(g(x) + 2)^2} \).

Step 1: Identify the Derivative Formula

To find the derivative of the function \( h(x) = \frac{3f(x)}{g(x) + 2} \), we will use the quotient rule:\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \] where \( u = 3f(x) \) and \( v = g(x) + 2 \).

Step 2: Differentiate the Numerator

Calculate the derivative of the numerator \( u = 3f(x) \). Since 3 is a constant multiplier, the derivative is \( u' = 3f'(x) \).

Step 3: Differentiate the Denominator

Calculate the derivative of the denominator \( v = g(x) + 2 \). The constant term 2 differentiates to 0, so the derivative is \( v' = g'(x) \).

Step 4: Apply the Quotient Rule

Substitute \( u, v, u' \), and \( v' \) into the quotient rule formula. We have:\[ h'(x) = \frac{(g(x) + 2) \cdot 3f'(x) - 3f(x) \cdot g'(x)}{(g(x) + 2)^2} \]

Step 5: Simplify the Expression

Simplify the expression if necessary. The derivative of \( h(x) \) is:\[ h'(x) = \frac{3(g(x) + 2)f'(x) - 3f(x)g'(x)}{(g(x) + 2)^2} \]

Key Concepts

Differentiable Functions

When we say that a function is differentiable, it means that the function has a derivative everywhere in its domain. This concept is foundational in calculus and is crucial when working with real-world applications.Differentiable functions:

• Are continuous everywhere in their domain. A jump or break means not differentiable.
• Have a defined slope or tangent at every point. This slope is what we call the derivative.

For function combinations like products, quotients, or sums, the differentiability must be checked individually and together. In our exercise, both functions, \( f(x) \) and \( g(x) \), are differentiable. This is important because the quotient rule, which we use to find the derivative of the given function, is applicable only when both numerator and denominator functions are differentiable.

Derivative of a Function

The derivative of a function represents its rate of change with respect to a variable. It's like measuring the speed of a car at any given moment. If \( y = f(x) \) is your function, its derivative \( f'(x) \) gives you how much \( y \) changes when \( x \) slightly changes. To find the derivative:

• Apply rules like power, product, or quotient rule depending on the function's form.
• Compute limits to ensure accurate rates of change.

For the function \( h(x) = \frac{3f(x)}{g(x) + 2} \), we use the quotient rule to determine its derivative. This rule is useful as it accounts for variations in both parts of the fraction. Using the rule, we differentiate both the top \( u \) and bottom \( v \) part of the fraction, something necessary for our function.

Numerator and Denominator Differentiation

In finding derivatives of quotients, handle the numerator and denominator differently but systematically. Start by identifying what your \( u \) and \( v \) represent in a quotient form \( \frac{u}{v} \) and differentiate each.**Differentiating the Numerator:**

• For \( u = 3f(x) \), bring out constants (here the 3), and multiply by the derivative of \( f(x) \), hence \( u' = 3f'(x) \).

**Differentiating the Denominator:**

• For \( v = g(x) + 2 \), the rule is simpler since constants vanish upon differentiation, leaving \( v' = g'(x) \).

Once both parts are differentiated, substitute back into the quotient rule formula. Proper differentiation of both parts is essential for accurate calculations, as seen in finding \( h'(x) \). This methodical differentiation reveals how the alteration in one function impacts the overall rate of change.