Applying derivative rules, it is found that p'(1) = 19.
The function given is:
[tex]p(x) = f(5x) + f(g(x)) + g(x)^2[/tex]
The derivative is the sum of derivatives, thus:
[tex]p^{\prime}(x) = [f(5x)]^{\prime} + [f(g(x))]^{\prime} + [g(x)^2]^{\prime}[/tex]
Applying the shifting rule:
[tex][f(5x)]^{\prime} = 5f^{\prime}(x)[/tex]
Applying the chain rule:
[tex][f(g(x))]^{\prime} = f^{\prime}(g(x))g^{\prime}(x)[/tex]
Applying the power of x, along with the chain rule:
[tex][g(x)^2]^{\prime} = 2g(x)g^{\prime}(x)[/tex]
Thus:
[tex]p^{\prime}(x) = 5f^{\prime}(x) + f^{\prime}(g(x))g^{\prime}(x) + 2g(x)g^{\prime}(x)[/tex]
At x = 1:
[tex]p^{\prime}(1) = 5f^{\prime}(1) + f^{\prime}(g(1))g^{\prime}(1) + 2g(1)g^{\prime}(1)[/tex]
From the table:
[tex]p^{\prime}(1) = 5(-1) + 6f^{\prime}(0)[/tex]
[tex]p^{\prime}(1) = -5 + 6(4)[/tex]
[tex]p^{\prime}(1) = 19[/tex]
A similar problem is given at https://brainly.com/question/22257133
