Problem 42 - Calculus an Applied Approach

Question

Find \(f^{\prime}(x)\) $$ f(x)=\left(3 x^{2}-5 x\right)\left(x^{2}+2\right) $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = (3x^2 -5x)(x^2+2)\) is \(f'(x) = 12x^3 - 15x^2 +12x -10\).

Step 1: Identify the First and the Second functions

Here, the first function can be taken as \(f_1(x) = 3x^2 - 5x\) and the second function can be taken as \(f_2(x) = x^2 + 2\). The final function \(f(x) = f_1(x) \cdot f_2(x)\) is the product of these two functions.

Step 2: Apply the Power Rule

Taking the derivative of \(f_1(x)\) using the power rule, we get \(f_1'(x) = 6x - 5\), and the derivative of \(f_2(x)\) gives \(f_2'(x) = 2x\).

Step 3: Apply the Product Rule

Now, apply the product rule. The derivative of a product of two functions (in this case \(f_1(x)\) and \(f_2(x)\)) is equivalent to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. Thus, \(f'(x) = f_1'(x) \cdot f_2(x) + f_1(x) \cdot f_2'(x)\). Replacing the values from Step 2 and Step 1, \(f'(x) = (6x-5)(x^2+2) + (3x^2-5x)(2x)\).

Step 4: Simplify

Now, simplify this equation to find the final form of the derivative. Thus, \(f'(x) = 6x^3 +12x - 5x^2 - 10 + 6x^3 -10x^2 = 12x^3 - 15x^2 +12x -10\).

Key Concepts

Power Rule

The Power Rule is a cornerstone of calculus and is vital for finding derivatives. It states that when you want to find the derivative of a function of the form \(x^n\), you simply bring the exponent \(n\) down as a coefficient and reduce the exponent by one. So, the derivative of \(x^n\) is \(n \cdot x^{n-1}\).
For example, in the function \(f_1(x) = 3x^2 - 5x\), applying the Power Rule to \(3x^2\) yields \(6x\) because 2 (the power) multiplied by 3 is 6, and \(x\) to the power of 1 remains \(x\). Similarly, for \(-5x\) where the power is 1, the derivative becomes \(-5\) after reducing the power by one.
This rule makes finding derivatives simpler, especially in polynomial expressions.

Derivative

A derivative, in simple terms, represents the rate at which a function is changing at any given point. It gives us information about the slope of the tangent line to the graph of the function at any point on the curve.
In essence, derivatives can help us understand how a small change in the input of a function (x-axis) affects the output of the function (y-axis).

• A positive derivative indicates that the function is increasing.
• A negative derivative shows the function is decreasing.
• A zero derivative illustrates that the function has a flat, horizontal tangent line.

When we compute the derivative of the function \(f(x)\), we denote it as \(f'(x)\). For the exercise provided, using the product and power rule, we computed \(f'(x)\) which represents the derivative of the entire function.

Differentiation

Differentiation is the process of finding a derivative. It involves computing the limit of the average rate of change of the function in question.
The differentiation process can be straightforward for simple polynomials using basic rules like the Power Rule. However, it can become more complex with functions formed by the product, quotient, or composition of different types of functions.
In our exercise, we used differentiation rules on both parts of the polynomial \((3x^2 - 5x)\) and \((x^2 + 2)\) to find their derivatives. These steps involve:

• Identifying the forms of the components, whether they are powers of x, constants, or other functions.
• Applying the differentiation rules systematically.
• Simplifying the resulting expressions to reach a final form.

With these steps, differentiation becomes a structured method for analyzing functions.

Polynomial

A polynomial is a mathematical expression consisting of variables and coefficients, involving the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Polynomials appear frequently in calculus problems and have a degree, which is the highest power of the variable present.
In the example exercise, both \(f_1(x) = 3x^2 - 5x\) and \(f_2(x) = x^2 + 2\) are polynomials where linear and quadratic terms are present. Polynomials of higher degree can be handled with similar derivative rules but may require more steps due to their complexity.
When we multiply two polynomials together, like in this exercise, the result is also a polynomial. The degree of this new polynomial will generally be the sum of the degrees of the original polynomials.