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. $$ \text {Function} \quad \text {Rewrite} \quad \text {Differentiate} \quad \text {Simplify} $$ \(y=\frac{5 x^{2}-3}{4}\)

Short answer

The derivative of the function \(y = \frac{5x^{2}-3}{4}\) is \(y' = \frac{5}{2}x\).

Step-by-step solution

Rewrite the function

The given function can be rewritten by dividing each term individually by the constant in the denominator. The revised function is \(y = \frac{5}{4}x^{2} - \frac{3}{4}\).

Differentiate

Now, differentiate each term individually with respect to \(x\): The derivative of the first term is done using the power rule and the constant multiplication rule. The derivative of \(x^{2}\) is \(2x\), and when multiplied by \(\frac{5}{4}\), the derivative of the first term becomes \(\frac{5}{2}x\). The second term, \(-\frac{3}{4}\), is a constant, so its derivative is 0.

Simplify

In this case, there is no need for simplification. The derivative stays as \(\frac{5}{2}x\).

Key concepts

Understanding the Power Rule

The power rule is a simple and essential tool in calculus for finding derivatives of polynomial functions. It states that if you have a function of the form \(f(x) = x^n\), where \(n\) is any real number, the derivative of this function is \(f'(x) = nx^{n-1}\). This rule provides a straightforward method to differentiate terms without needing complex calculations.

For example, if you have \(f(x) = x^2\), using the power rule, you determine the derivative as follows:

• Apply the exponent to the coefficient (here 1, implied) to get \(2\times1\).
• Subtract one from the exponent, resulting in the expression \(2x^{1}\).

The result, \(2x\), is the derivative of \(x^2\) with respect to \(x\). Using this technique, complex polynomial expressions turn into easily manageable forms, simplifying the process of differentiation considerably.

Applying the Constant Multiple Rule

The constant multiple rule is an extremely practical differentiation technique that simplifies the process when a constant is involved in a term. It states that if a function \(f(x) = c \, g(x)\), where \(c\) is a constant and \(g(x)\) is a differentiable function, the derivative \(f'(x)\) is simply \(c\, g'(x)\).

This means you can "factor out" the constant and focus on differentiating \(g(x)\). For example, consider the function \(y = \frac{5}{4}x^2\). Here, \(\frac{5}{4}\) is a constant, so using the constant multiple rule:

• Leave the \(\frac{5}{4}\) as is.
• Differentiate \(x^2\) as usual with the power rule, which gives \(2x\).
• Multiply the derivative \(2x\) by the constant \(\frac{5}{4}\) to obtain \(\frac{5}{2}x\).

This allows for a more intuitive and less error-prone method to calculate derivatives when constants are present.

Differentiation Techniques Simplified

Differentiation is a core concept in calculus, used to find the rate at which a function is changing at any given point. Understanding the variety of techniques available ensures that you can approach any problem confidently. The balance comes from knowing when to use specific rules, such as the power rule or constant multiple rule.

When differentiating functions, breaking the problem down into manageable parts is crucial. Consider a function composed of multiple terms or constants, as in the exercise function \(y = \frac{5}{4}x^2 - \frac{3}{4}\). By applying appropriate rules:

• Utilize the power rule for polynomial terms, transforming them into their derivative forms easily.
• Employ the constant multiple rule for constant coefficients, simplifying by "factoring" them out of the differentiation process.
• Keep constant terms in mind; their derivatives are zero, emphasizing the importance of correctly applying rules.

Mastering these basic techniques with practice will make the more complex aspects of calculus easier to tackle, laying a solid foundation for advanced topics.