Problem 36 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$h(x)=\frac{a x+b}{c}$$

Step-by-Step Solution

Answer

The derivative of \(h(x) = \frac{a x + b}{c}\) is \(\frac{a}{c}\).

Step 1: Identify the function

The function given is \( h(x) = \frac{a x + b}{c} \). This is a linear function divided by a constant, \( c \).

Step 2: Apply the derivative formula

To find the derivative of a function \( h(x) = \frac{u}{v} \), where \( u = a x + b \) and \( v = c \), we apply the quotient rule. However, since \( v \) is a constant, we can simplify it by directly taking the derivative of the numerator \( u \) and then dividing by \( c \).

Step 3: Differentiate the numerator

The derivative of the numerator \( u = a x + b \) with respect to \( x \) is \( a \), because the derivative of \( ax \) is \( a \) and the derivative of a constant \( b \) is 0.

Step 4: Express the derivative

The derivative of \( h(x) \) is the derivative of the numerator \( u \) divided by the constant \( c \). Thus, \( h'(x) = \frac{a}{c} \).

Key Concepts

Linear Function

A linear function is a function that creates a straight line when graphed. It can typically be expressed in the form \( f(x) = a x + b \), where:

• \(a\) is the slope of the line, describing how steep the line is.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

Linear functions are simple yet powerful because of their consistent rate of change, which is determined by the slope \(a\).
In terms of calculus, whenever you see a function of the type \(ax + b\), it's instantly identifiable as a linear function. It plays a major role in the derivatives since their simplicity makes calculation rather straightforward. The given problem, \(h(x) = \frac{ax + b}{c}\), shows a linear function, \(ax + b\), divided by a constant, \(c\). This division by a constant doesn't affect the linear nature but influences calculations such as differentiation.

Constant Differentiation

The concept of constant differentiation is key when dealing with derivatives of linear functions divided by a constant. In any function, constants reveal certain properties:

• When differentiating a constant alone, the result is always 0.
• If a constant is multiplying a function, this constant factor remains as it is during differentiation.

In our case, we are given a linear function, \(ax + b\), divided by a constant \(c\). This means when you find the derivative of \(ax + b\), it becomes \(a\). Since you're dividing by \(c\), the derivative of the whole expression is \(\frac{a}{c}\).
Understanding these rules makes differentiating functions that have constants straightforward and quick.

Quotient Rule

The quotient rule is a method used to find the derivative of a function that is the quotient of two other functions. For functions of the form \(g(x) = \frac{u}{v}\), where both \(u\) and \(v\) are differentiable, you typically apply the rule:\[g'(x) = \frac{v \cdot u' - u \cdot v'}{v^2}\]In our specific problem, examining \(h(x) = \frac{ax + b}{c}\), notice that \(v=c\) is a constant. This simplifies matters because the derivative of a constant, \(v'\), is 0. Thus, the quotient rule simplifies, allowing us to find the derivative by just differentiating the numerator and dividing by the constant.Key Simplification: Because \(v\) is constant, the challenging parts of quotient rule shrink. So, for \(h(x)\), we directly find the result by dividing \(u' = a\) by \(c\). The quotient rule does not need traditional deployment, making deriving such expressions quicker.