Problem 7 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=3 t^{4}-2 t^{2}$$

Step-by-Step Solution

Answer

The derivative is \( y' = 12t^3 - 4t \).

Step 1: Apply the Power Rule to Each Term

To find the derivative of the function \( y = 3t^4 - 2t^2 \), we apply the power rule separately to each term. The power rule states that the derivative of \( t^n \) is \( n imes t^{n-1} \).

Step 2: Differentiate the First Term

Differentiate the first term \( 3t^4 \). According to the power rule, \( \frac{d}{dt}(3t^4) = 3 \times 4t^{4-1} = 12t^3 \).

Step 3: Differentiate the Second Term

Differentiate the second term \( -2t^2 \). Using the power rule, \( \frac{d}{dt}(-2t^2) = -2 \times 2t^{2-1} = -4t \).

Step 4: Combine the Derivatives

Combine the results from Step 2 and Step 3 to write the final derivative of the function. The derivative \( y' = 12t^3 - 4t \).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus used to find the derivative of power functions. It simplifies the process of calculating derivatives by providing a straightforward formula. When using the power rule, if you have a term in the form of \( t^n \), the derivative is given by \( n \times t^{n-1} \). This rule allows you to quickly differentiate any power of \( t \):

• Raise the exponent to the coefficient by multiplying \( n \) with the original coefficient.
• Decrease the exponent by one.

This method is particularly useful when dealing with polynomial expressions. Given a function, you can apply the power rule term by term, making the overall differentiation process both manageable and efficient.

Differentiation

Differentiation is a core concept in calculus, representing the process of finding the derivative of a function. A derivative gives the rate at which a function is changing at any given point and is crucial in understanding motion and change.
This process involves applying rules, like the power rule, to compute derivatives of functions. During differentiation:

• You evaluate the slope of the tangent line to the function at any point.
• You reveal insights into the function's behavior, such as increase or decrease.

Differentiation helps graph functions more accurately and is essential in optimizing real-world problems, from engineering to economics.

Calculus

Calculus is the mathematical study of continuous change. It has two main branches: differentiation and integration. While differentiation focuses on the rate of change and slopes of curves, integration is concerned with areas under curves and cumulative totals. Calculus provides a framework for modeling dynamic systems, and understanding calculus concepts can lead to more profound insights into natural and technological phenomena.
Basic calculus skills, like using the power rule for differentiation, form the foundation for more advanced topics in mathematics and science. With its wide applications, calculus is widely regarded as a gateway to advanced fields of study, providing essential tools for analyzing change and understanding relationships between variables in a variety of contexts.