Problem 32 - Calculus an Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ g(x)=\sqrt{5-3 x} $$

Step-by-Step Solution

Answer

The derivative of the function \(g(x) = \sqrt{5 - 3x}\) is \(g'(x) = - (3 / 2\sqrt{5 - 3x})\).

Step 1: Express the Function as a Power of x

Rewrite the function \(g(x) = \sqrt{5 - 3x}\) as \(g(x) = (5 - 3x)^{1/2}\). This way, it aligns better with the general power rule. Here \(u = 5 - 3x\) and \(n = 1/2\).

Step 2: Apply the General Power Rule

Using the general power rule, \(d/dx(u^n) = nu^{n-1}u'\), apply it to the function. But first find \(u'\), the derivative of \(u = 5 - 3x\). The derivative \(u' = -3\). Now apply the general power rule which yields \(1/2*(5 - 3x)^{(1/2)-1}*(-3)\).

Step 3: Simplify the Expression

Simplify the expression obtained in the previous step. The expression then becomes \(-(3/2)(5 - 3x)^{-1/2}\).

Step 4: Return to Original Form

The last step involves rewriting the derivative back to the radical form since the original function was given in that form. The derivative \(g'(x)\) becomes \(- (3 / 2\sqrt{5 - 3x})\)

Key Concepts

Derivative

The derivative of a function is a fundamental concept in calculus, representing the rate at which the function's value changes as its input changes.
For any given function, the derivative tells us how steep the function's graph is at any point along its curve. This provides valuable insights into the behavior of the function, such as its increasing or decreasing trends and any points where it flattens or turns around.

• In calculus, the notation for the derivative of a function is typically written as \( f'(x) \) or \( \frac{df}{dx} \).
• Finding a derivative involves applying specific rules and techniques based on the form of the function.

For example, to find the derivative of \( g(x) = \sqrt{5-3x} \), you would rewrite the square root as an exponent to make it possible to use the General Power Rule, thus finding the rate of change of \( g(x) \) with respect to \( x \).

Function Notation

Function notation is a way to denote functions in a compact and clear manner.
In functions, we often use symbols such as \( f(x) \) to describe the function's rule, in terms of the variable \( x \).

• For instance, in the given function \( g(x) = \sqrt{5-3x} \), "\( g(x) \)" simply refers to the function named "g" evaluated at the input "x".
• This way of writing functions is useful because it instantly communicates both the function itself and the variable it's being evaluated with.

Using function notation makes it easy to differentiate between different functions and keep track of which variable you are differentiating with respect to when taking calculations like derivatives.

Exponent Rules

Understanding exponent rules is key to rewriting and differentiating functions, especially when they involve roots or powers.
Exponents are a shorthand way to represent repeated multiplication or division. Some of the fundamental rules include:

• \((a^m)(a^n) = a^{m+n}\) - multiplying like bases adds their exponents.
• \((a^m)/(a^n) = a^{m-n}\) - dividing like bases subtracts the exponents.
• \((a^m)^n = a^{m*n}\) - raising a power to a power multiplies the exponents.

When the exercise rewrites \( g(x) = \sqrt{5-3x} \) as \( (5-3x)^{1/2} \), it applies the rule that a square root can be expressed as a half exponent, which then allows the application of the General Power Rule to find the derivative.

Radical Expression

A radical expression includes roots such as square roots, cube roots, etc.
Radicals often appear in different forms in mathematics, and they're closely related to exponentiation, where, for example, the square root of a number can be represented as that number raised to the power of \( 1/2 \).

• Radicals can complicate expressions unless converted to exponents for ease of differentiation or integration.
• In the context of calculus and finding derivatives, radicals are commonly rewritten using exponent rules.

In the problem \( g(x) = \sqrt{5 - 3x} \), rewriting the radical as \( (5 - 3x)^{1/2} \) allows us to use standard rules for derivatives, like the General Power Rule, to compute the rate of change efficiently.