Final answer:
The derivative of h(x) = 5f(x) - 3gx is obtained by applying the sum rule and constant multiple rule, resulting in 5f'(x) - 3g'(x).
Step-by-step explanation:
The student's question asks how to find the derivative of the function h(x) = 5f(x) - 3gx. To find the derivative, we apply the sum rule and the constant multiple rule in differentiation.
The sum rule states that the derivative of a sum of functions is the sum of their derivatives, and the constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function. Therefore, the derivative of h(x) is the derivative of 5f(x) minus the derivative of 3gx, which is 5f'(x) - 3g'(x).
To find the derivative of h(x) = 5f(x) - 3gx with respect to x, apply the rules of differentiation. The derivative of 5f(x) is 5 times the derivative of f(x), and the derivative of -3gx is -3 times the derivative of gx. Expressing it concisely, h'(x) = 5f'(x) - 3g'(x). This represents the rate of change of h(x) concerning x, incorporating the derivatives of the functions f(x) and g(x). The product rule and constant factor rule for derivatives are applied here, facilitating the process of finding the derivative of the given composite function.