Question

Using the Constant Multiple Rule In Exercises 19-24, complete the table to find the derivative of the function without using the Quotient Rule. $$ \text {Function} \quad \text {Rewrite} \quad \text {Differentiate} \quad \text {Simplify} $$ \(y=\frac{4 x^{3 / 2}}{x}\)

Short answer

The derivative of the function \(y=\frac{4 x^{3 / 2}}{x}\) is \(\frac{2}{x^{1/2}}\)

Step-by-step solution

Simplification

Simplify the function. The fraction can be reduced by dividing \(x^{3 / 2}\) by \(x\) to get \(x^{1 / 2}\). Hence, rewrite the function as \(y = 4x^{1 / 2}\)

Differentiation

Differentiate the simplified function. We can use the power rule here, which states that the derivative of \(x^n\), where 'n' is any real number, is \(n*x^{n-1}\). Applying this rule to our function, we get \(\frac{dy}{dx} = 4 * 1/2 * x^{1 / 2 - 1}\)

Simplification of derivative

Now simplify the derivative. It becomes \(\frac{dy}{dx} = 2*x^{-1/2}\)

Rewrite to Positive Exponent

We typically want to avoid negative exponents in our final answer. Convert the negative exponent to a positive exponent by writing it in the denominator: \(\frac{dy}{dx} = \frac{2}{x^{1/2}}\).

Key concepts

Power Rule

The power rule is one of the simplest and most commonly used techniques in calculus to find the derivative of a function. It applies to terms in the form of \(x^n\), where "n" is a real number. To use the power rule, simply multiply the exponent by the coefficient in front of \(x\), and then subtract one from the exponent. This gives you the derivative:

• Original form: \(x^n\)
• Derivative: \(n*x^{n-1}\)

Let's consider an example where the function is \(x^{3/2}\). By applying the power rule, the derivative becomes \(\frac{3}{2}x^{3/2-1}\) or \(\frac{3}{2}x^{1/2}\). This rule makes differentiation straightforward for polynomials and variable expressions.

Constant Multiple Rule

The constant multiple rule is an invaluable tool for simplifying the differentiation process. It states that a constant multiplied by a function can be factored out, simplifying the differentiation process. This allows you to focus on differentiating the function itself without worrying about the constant. The rule can be summarized as:

• Original form: \(a*f(x)\)
• Derivative: \(a\cdot\frac{d}{dx}f(x)\)

Take, for example, the function \(y = 4x^{1/2}\). Here, the 4 is a constant. By applying the constant multiple rule, you can factor out the 4 and differentiate only the \(x^{1/2}\). The result would be \(4 \cdot \frac{1}{2}x^{-1/2}\), leading to a final derivative of \(2x^{-1/2}\). This rule simplifies calculations and reduces the chance of error.

Simplification

Simplification in calculus often involves rewriting expressions to make them easier to work with, both algebraically and during differentiation. Simplifying the function by dividing coefficients and combining like terms is a typical first step.For instance, we started with the function \(\frac{4x^{3/2}}{x}\). By simplifying, we divided \(x^{3/2}\) by \(x\) which can be rewritten using exponent subtraction: \(x^{3/2 - 1} = x^{1/2}\). This leads to the simplified function \(y = 4x^{1/2}\).Simplification is crucial because it not only makes differentiating easier but also helps to avoid complex terms and cumbersome calculations in subsequent steps.

Negative Exponents

Dealing with negative exponents is a common aspect of calculus, especially when differentiating or simplifying functions. A negative exponent indicates that the base is on the wrong side of the fraction line; in other words, it should be in the denominator.For example, in differentiating \(y = 4x^{1/2}\), we found the derivative \(\frac{dy}{dx} = 2x^{-1/2}\). To express this without negative exponents, rewrite \(x^{-1/2}\) as \(\frac{1}{x^{1/2}}\). Consequently, the derivative becomes \(\frac{2}{x^{1/2}}\).Handling negative exponents correctly is essential for final answers, as it often leads to a more simplified and accepted mathematical expression. This positive exponent form is generally preferred in most contexts.