Final answer:
The question focuses on differentiation using the power rule and product rule. The power rule is applied to each term of a polynomial separately. Calculating instantaneous velocity and handling reciprocal exponents are also briefly explained.
Step-by-step explanation:
Understanding the Power Rule in Differentiation
To differentiate functions involving powers of x, we use the power rule of differentiation. This rule states that d/dx[xn] = nxn-1, where n is a real number. When differentiating a function like f(x) = 7x5 − x9, we apply the power rule to each term separately. For the first term, 7x5, the derivative is 35x4, as we multiply the exponent by the coefficient and decrease the exponent by one. The product rule is used when differentiating products of functions. For instance, if we have a function g(x) = f(x)h(x), the derivative g'(x) = f'(x)h(x) + f(x)h'(x).
Calculating instantaneous velocity often requires differentiating position functions of time, x(t), using the power rule. If the position function terms have the form Atn, their derivatives can be found analogously. In more complex cases, understanding the inverse relationship between operations is essential. An example is the Pythagorean Theorem, which is used to calculate the side of a right triangle by 'undoing' the square through taking a square root.
Equations with negative exponents represent division operations. As shown in Equation A.9, x−n = 1/xn, which expresses a negative exponent as a reciprocal.
Example of Power Rule
To illustrate the power rule, consider f(x) = x2. Its derivative is f'(x) = 2x2-1 or simply 2x. In a real-world context, if x2 represents the area of a square, the derivative provides us with the function that tells us how quickly the area changes with respect to one side of the square.