Final answer:
The correct answer is option b. The derivative of g(x) = 7x² is 14x. The derivative of h(x) = x - 2 is 1. The derivative of f(x) = 7x²(x - 2) is 21x².
Step-by-step explanation:
The derivative of g(x) = 7x² is obtained by applying the power rule of differentiation. According to the power rule, the derivative of x^n, where n is a constant, is given by nx^(n-1).
Applying this rule to g(x), we have:
- g'(x) = 2(7x)^(2-1) = 14x
Therefore, the derivative of g(x) = 7x² is g'(x) = 14x.
For h(x) = x - 2, the derivative can be found using the constant rule and the power rule. The constant rule states that the derivative of a constant multiplied by a function is equal to the constant times the derivative of the function. Since -2 is a constant, we have:
Finally, for f(x) = 7x²(x - 2), we can find the derivative by applying the product rule. The product rule states that the derivative of the product of two functions u(x) and v(x) is equal to u'(x)v(x) + u(x)v'(x). Applying this rule to f(x), we get:
- f'(x) = 7x²(1) + 2(7x²) = 7x² + 14x² = 21x²
Therefore, the derivative of f(x) = 7x²(x - 2) is f'(x) = 21x².