Problem 33 - Single Variable Calculus

Question

Find the derivative of the function. Let \(f(x)=2 x^{3}-4 x\), Find a. \(f^{\prime}(-2)\) b. \(f^{\prime}(0)\) c. \(f^{\prime}(2)\)

Step-by-Step Solution

Answer

a. \(f^{\prime}(-2) = 20\) b. \(f^{\prime}(0) = -4\) c. \(f^{\prime}(2) = 20\)

Step 1: Find the derivative of f(x)

To find the derivative of the given function, \(f(x) = 2x^3 - 4x\), we will apply the power rule of differentiation: \[\frac{d}{dx}(x^n) = nx^{n-1}\] Here, using the power rule, we differentiate each term in the function with respect to 'x'. \[\frac{d}{dx}(2x^3) = 6x^2\] \[\frac{d}{dx}(-4x) = -4\] The derivative of the function f(x) is: \[f'(x) = 6x^2 - 4\]

Step 2: Evaluate f'(x) at x = -2

Now that we have the derivative function, \(f'(x) = 6x^2 - 4\), we can evaluate it at the required points. First, we evaluate f'(x) at x = -2: \[f'(-2) = 6(-2)^2 - 4\] \[f'(-2) = 6(4) - 4\] \[f'(-2) = 24 - 4\] \[f'(-2) = 20\]

Step 3: Evaluate f'(x) at x = 0

Next, we evaluate f'(x) at x = 0: \[f'(0) = 6(0)^2 - 4\] \[f'(0) = 0 - 4\] \[f'(0) = -4\]

Step 4: Evaluate f'(x) at x = 2

Finally, we evaluate f'(x) at x = 2: \[f'(2) = 6(2)^2 - 4\] \[f'(2) = 6(4) - 4\] \[f'(2) = 24 - 4\] \[f'(2) = 20\] In conclusion, we find that: a. \(f^{\prime}(-2) = 20\) b. \(f^{\prime}(0) = -4\) c. \(f^{\prime}(2) = 20\)

Key Concepts

Power Rule of Differentiation

Understanding the power rule is a key to mastering calculus problems involving derivatives. Simplified, the power rule tells us that to differentiate a function of the form ax^n, where a is a constant and n is a real number, we multiply the exponent n by the coefficient a and then decrease the exponent by one. The mathematical representation of the power rule is:
\[\begin{equation}\frac{d}{dx}(ax^n) = anx^{n-1}\end{equation}\]
In the exercise, you apply this to each term of the function independently, such as 2x^3 and -4x. This approach simplifies the process of finding derivatives, which is essential when dealing with complex functions in calculus problems.

Derivative Evaluation

Once the derivative of a function is known, evaluating that derivative at specific points gives us the slope of the tangent line to the function at those points. This process is known as derivative evaluation. In the given example, after applying the power rule, the derivative of the function f(x) is found to be f'(x) = 6x^2 - 4. To evaluate this derivative, simply substitute the value of x with the numbers given, such as -2, 0, and 2, and then compute the result for each case.
These evaluations provide critical information, such as the rate of change of the function at specific points, which can be crucial for understanding the behavior of the function in calculus problems.

Calculus Problems

Calculus problems, like finding derivatives, are fundamental in understanding changes and motion. They frequently require analyzing functions, evaluating slopes, and estimating areas under curves. When you engage with calculus exercises, you're developing analytical thinking by linking the abstract concept of the derivative with tangible changes in variables represented by a function. This not only assists with academic pursuits but also has practical applications in physics, engineering, economics, and beyond. Applying the power rule and evaluating derivatives are tasks that serve as building blocks for more complex calculus problems, such as optimization and integration.

Function Differentiation

Differentiating a function is all about finding the derivative, which represents the rate at which the function's output value changes as its input value changes. It's a critical operation in calculus that gives insight into a function's growth or decay, maxima and minima, and general behavior. In our exercise, the function f(x) = 2x^3 - 4x is differentiated using the power rule, reflecting how straightforward function differentiation can be when understood properly. Differentiation helps to carve a clear path in the exploration of more advanced calculus problems, where functions can describe anything from simple motion to complex dynamic systems.