The value of h'(3) = 16, using differentiation and chain rule.
What is the chain rule?
According to the chain rule, if f and g are both differentiable and F = f.g is the composite function defined by F(x) = f(g(x)), then F is differentiable and F' is given as:
F'(x) = f'(g(x)).g'(x).
How to solve the question?
In the question, we are given that h(x) = 5 + 4f(x), and are asked to find the value of h'(3), when f(3) = 5, and f'(3) = 4.
We first differentiate the function h(x) = 5 + 4f(x), in terms of x applying the chain rule as follows:
[d/dx](h(x)) = [d/dx]{5 + 4f(x)},
or, h'(x) = [d/dx](5) + [d/dx]{4f(x)},
or, h'(x) = 4[d/dx]{f(x)},
or, h'(x) = 4f'(x).
As we are asked to find the value of h'(3), we substitute x = 3, in the above equation to get:
h'(3) = 4f'(3),
or, h'(3) = 4(4) {Since f'(3) = 4},
or, h'(3) = 16.
Thus, the value of h'(3) = 16, using differentiation and chain rule.
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