Question

Compute the derivative of the given function. $$m(t)=9 t^{5}-\frac{1}{8} t^{3}+3 t-8$$

Short answer

The derivative of \( m(t) \) is \( 45t^4 - \frac{3}{8}t^2 + 3 \).

Step-by-step solution

Understand the Problem

We need to find the derivative of the function given by \( m(t) = 9t^5 - \frac{1}{8}t^3 + 3t - 8 \). The derivative of a function represents the rate of change of the function with respect to its independent variable, \( t \), in this case.

Differentiate Each Term Separately

Differentiate each term of the function \( m(t) \) with respect to \( t \). Use the power rule for differentiation, which states that if \( f(t) = at^n \), then \( f'(t) = ant^{n-1} \).

Differentiate the First Term

Using the power rule, differentiate \( 9t^5 \). The derivative is \( 5 \times 9t^{5-1} = 45t^4 \).

Differentiate the Second Term

Differentiate \(-\frac{1}{8}t^3\) using the power rule. The derivative is \( 3 \times -\frac{1}{8}t^{3-1} = -\frac{3}{8}t^2 \).

Differentiate the Third Term

Differentiate \( 3t \). The derivative is \( 1 \times 3t^{1-1} = 3 \).

Differentiate the Constant Term

The derivative of a constant \( -8 \) is 0, because constants do not change with respect to \( t \).

Combine All Derivatives

Combine the derivatives of each term: \( 45t^4 - \frac{3}{8}t^2 + 3 \). This is the derivative of the function \( m(t) \).

Key concepts

Power Rule

When working with derivatives, the power rule is a fundamental tool that simplifies the process of differentiation. The rule applies to functions of the form \( f(t) = at^n \), where \( a \) is a constant coefficient and \( n \) is a real number exponent.

The power rule states that the derivative of \( f(t) \) is given by:
\[ f'(t) = ant^{n-1} \]
This means you multiply the original exponent \( n \) by the coefficient \( a \) to get the new coefficient and reduce the exponent by one.

For example, if you have a term like \( 9t^5 \), applying the power rule gives:

• The original coefficient \( 9 \) is multiplied by the original exponent \( 5 \), resulting in \( 45 \).
• The exponent is decremented by one, turning it from \( 5 \) into \( 4 \), yielding the term \( 45t^4 \).

Function Differentiation

Function differentiation refers to the process of finding the derivative of a function. This technique involves decomposing complex mathematical expressions into simpler terms and applying rules to find their respective derivatives. It's like peeling an onion, getting down to the core of each term.

With polynomials, such as our function \( m(t) = 9t^5 - \frac{1}{8}t^3 + 3t - 8 \), differentiation becomes a matter of breaking the expression into individual terms:

• Each term is differentiated separately according to its form.
• Use the power rule for terms like \( 9t^5 \) and \(-\frac{1}{8}t^3 \).
• Constant terms like \( 3t \) are differentiated by simplifying \( t^1 \) using the power rule.
• Pure constants such as \(-8\) yield a derivative of zero.

It's helpful to tackle derivatives one term at a time to prevent mistakes and ensure clarity.

Rate of Change

The derivative of a function is a powerful form of mathematical analysis used to describe its rate of change. It's essentially how we quantify the relationship between changes in one variable with respect to another.

For a function \( m(t) = 9t^5 - \frac{1}{8}t^3 + 3t - 8 \), finding its derivative \( m'(t) \) gives a new function that describes the instantaneous rate of change of \( m \) with respect to \( t \).

This rate of change is useful in many real-world scenarios, such as:

• Determining the speed of a moving object at any given time.
• Understanding acceleration or deceleration by observing changes in velocity over time.
• Modeling economic data to find growth rates or trends.

By simplifying the function to its derivative, you extract meaningful information that helps predict and analyze behavior across different contexts.