Question

The graph of functiog 9 and a table of values for function f and its derivatives are shown below. That is, for x=1,f(1)−2,f(i)=3,f∗(1)=−1 ote. (a) (i) Evaluate h′(3) when h(x)=g(r(x)) 9ⁿ (i) Evaluate h′(1) when h(x)=f(x)/g(x)​ (b) Evaluate h′′(2) when h′(x)=f′(x)×g′(x).

Answer 1

The question involves calculus, specifically the application of the chain rule, quotient rule, and product rule to compute the first and second derivatives of composite functions.

Step-by-step explanation:

The question revolves around the calculation of derivatives for a composite function h(x), and involves the application of the chain rule and quotient rule in calculus. For instance, when h(x) = g(f(x)), the derivative h'(x) is found by applying the chain rule: h'(x) = g'(f(x)) × f'(x). If h(x) = f(x)/g(x), the quotient rule is applied: h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2.

For the second derivative h''(x) where h'(x) = f'(x) × g'(x), product rule and differentiation are required once more to find the derivative of the product of two functions.