Question

Other Symbols for the Derivative If \(f(x)=7-4 x^{2},\) find \(f^{\prime}(x)\)

Short answer

\(f^{\rm \textprime}(x) = -8x\)

Step-by-step solution

Identify the function

Start by identifying the function to be differentiated which is given by the equation: \(f(x)=7-4 x^{2}\).

Apply the power rule for derivatives

To find the derivative of the given function, apply the power rule which states that the derivative of \(x^n\) with respect to \(x\) is \(n \times x^{n-1}\). Here you want to differentiate each term of the function separately.

Differentiate the constant term

The derivative of any constant like 7 (regardless of its value) with respect to \(x\) is 0. So the derivative of the first term in the function \(f(x)\) is 0.

Differentiate the quadratic term

Using the power rule, the derivative of \(-4x^2\) with respect to \(x\) is \(-4 \times 2 \times x^{2-1}\) which simplifies to \(-8x\).

Combine the derivatives

Combine the derivatives of the terms to obtain the final derivative. Since the derivative of the constant term is 0, it disappears in the final expression of the derivative, leaving you with just the derivative of the quadratic term.

Key concepts

Power Rule

The power rule is an essential concept in calculus, particularly when it comes to differentiation. It states that if you have a function in the form of x^n, where n is any real number, the derivative of that function with respect to x is n*x^(n-1). To apply the power rule, you just need to multiply the exponent n by the base x and then subtract one from the exponent.

This rule drastically simplifies the process of differentiation, especially for polynomial functions, as it lets you easily find the derivative of each term individually. For example, if you have the term 5x^3, applying the power rule gives you the derivative 3*5x^(3-1) = 15x^2. Remember, the power rule applies to any term with a power of x, making it a quintessential tool for anyone learning differentiation.

Derivative of a Function

Understanding the derivative of a function is fundamental in calculus. A derivative represents the rate at which a function is changing at any point. Formally, it gives us the slope of the tangent line to the function's graph at a particular point.

When we express f'(x) or df/dx, we're referring to the derivative of a function f(x) with respect to x. Calculating this value can tell us, for instance, the velocity of an object if f(x) represents its position with respect to time. Differentiation, the process of finding the derivative, involves applying rules like the power rule, as well as the product rule, quotient rule, and chain rule, depending on the complexity of the function.

Constant Term Differentiation

In differentiation, when dealing with a constant term—in other words, a number that does not change and has no variable attached to it—the derivative is always zero. That's because differentiation measures change, and a constant doesn't change.

For instance, if f(x) = 7, there's no x for the function to vary by, so no matter how much x changes, the value of f(x) doesn't budge. Hence, the derivative f'(x) = 0. This is a simple yet powerful concept, as it tells us that constant terms can be ignored when differentiating a function since they contribute nothing to the rate of change.

Quadratic Term Differentiation

Quadratic terms are found in functions where the variable is raised to the second power, like x^2. To differentiate a quadratic term, we use the power rule. Since the exponent is 2, we would multiply the term by 2 and then reduce the exponent by 1.

For example, if our function has a term -4x^2, applying the power rule gives us a derivative of -4*2*x^(2-1), which simplifies to -8x. It's important in the differentiation of quadratic terms to not only focus on the exponent but also to remember to carry any coefficients through the differentiation process just like we did with -4 in this example.