Product and quotient rule

Question

In the function h(x) below, defined in terms of f(x) and g(x), determine h′(2) in each case if f(2) = 3, g(2) = 4, f′(2) = 1, and g′(2) = −5.

1. h(x)=g(x)/f(x)+2

Answer 1

As mentioned in a comment on this question, there is some ambiguity regarding the grouping of the right-hand side of the equation. If the "+ 2" is not part of the denominator, then it would simply become 0 when you find the derivative of h(x). In that case, using the quotient rule, we would then have

h'(x) = [g'(x)*f(x) - g(x)*f'(x)] / [f(x)]².

From there, simply replace each quantity with its given value and simplify.

However, if the "+ 2" IS included in the denominator, this would change. The derivative would then become

h'(x) = [g'(x)*(f(x) + 2) - g(x)*f'(x)] / [f(x) + 2]².

And again, replace with the given values and simplify accordingly.

In either case, the problem should not be too challenging. Just an exercise in using the quotient rule to differentiate a function, and then find the gradient of the function at a specific value of x given certain preliminary values and information.