Problem 9 - Calculus and its applications

Question

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(f(x)=(2 x+5)\left(3 x^{2}-4 x+1\right)\)

Step-by-Step Solution

Answer

The derivative using both methods is the same: \( f'(x) = 18x^2 + 14x - 18 \).

Step 1: Differentiate using the Product Rule

The Product Rule states that if you have two functions that are being multiplied, say \( u(x) \) and \( v(x) \), their derivative is given by \( u'(x)v(x) + u(x)v'(x) \). Here, let \( u(x) = (2x+5) \) and \( v(x) = (3x^2 - 4x + 1) \). First, find the derivatives: \( u'(x) = 2 \) and \( v'(x) = 6x - 4 \). Then apply the Product Rule:\[f'(x) = (2)(3x^2 - 4x + 1) + (2x+5)(6x - 4) \].

Step 2: Simplify the Product Rule Result

To simplify \( f'(x) \), distribute in each group:\( (2)(3x^2 - 4x + 1) \) becomes \( 6x^2 - 8x + 2 \).\( (2x+5)(6x - 4) \) becomes \( 12x^2 - 8x + 30x - 20 = 12x^2 + 22x - 20 \).Add these results:\[f'(x) = (6x^2 - 8x + 2) + (12x^2 + 22x - 20) = 18x^2 + 14x - 18\].

Step 3: Multiply Expressions Before Differentiating

Multiply \( (2x + 5)(3x^2 - 4x + 1) \) before differentiating. Distribute each term:\[ 2x(3x^2 - 4x + 1) = 6x^3 - 8x^2 + 2x \]\[ 5(3x^2 - 4x + 1) = 15x^2 - 20x + 5 \]Combine these results:\[ f(x) = 6x^3 + 7x^2 - 18x + 5 \].

Step 4: Differentiate the Combined Expression

Now differentiate the simplified expression \( f(x) = 6x^3 + 7x^2 - 18x + 5 \). Using power rule:\[ f'(x) = 18x^2 + 14x - 18 \].

Step 5: Compare Results and Verify Using Graphing Calculator

Both methods yield the derivative \( f'(x) = 18x^2 + 14x - 18 \). They match, confirming consistency. Use a graphing calculator's derivative feature to verify: check if graphing the derivative and deriving it with calculator functions yields the same expression of \( 18x^2 + 14x - 18 \). The results match, validating our earlier calculations.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus used to determine the rate of change of a function. By finding the derivative of a function, you can understand how the function behaves as the input changes. In this exercise, we deal with differentiating a function composed of two expressions multiplied together:

• First, we apply the Product Rule, which is specifically designed for functions that are products of two or more terms.
• Next, we explored a different method by simplifying the function first and then differentiating.

Both methods served to double-check the accuracy of our calculations. Differentiation, in essence, helps us find slopes of tangent lines, optimize functions, and solve real-world problems involving rates of change and motion. Understanding how to differentiate functions and employing different rules, like the Product Rule, is essential for solving many calculus problems effectively.

Graphing Calculator Verification

Using a graphing calculator aids tremendously in verifying calculus problems, such as differentiating functions. Once we have derived the function by different methods, a graphing calculator helps confirm results by plotting the original and the derived functions, allowing us to visualize the slope and tangent changes. A graphing calculator can:

• Plot the function to provide a graphical representation of the derivative.
• Uses built-in functions to calculate the derivative symbolically or numerically with accuracy.

By comparing the calculated and plotted derivative with our manual results, the calculator acts as a dependable check. It ensures our manual differentiation calculations are correct, by revealing any discrepancies if the manual results don’t match the calculator’s output. This tool is beneficial to visually confirm our algebraic steps and explore any errors in calculations.

Power Rule

The Power Rule is a basic yet vital technique in differentiation, determining the derivative of functions in the form of powers of a variable. It's a straightforward concept where you bring down the exponent as a coefficient and reduce the exponent by one. For example, the derivative of a function \( x^n \) is \( n \, x^{n-1} \). In the exercise:

• The Power Rule was applied after expanding the expression and re-combining the individual derivatives.
• Using the Power Rule makes differentiation simpler and is a preferred method when dealing with polynomial expressions.

It’s essential to grasp the Power Rule as it provides a powerful tool for quickly finding slopes and rates of change. When you have expressions with variables raised to a power, this rule provides a quick and reliable method to differentiate the function.