"Let F(1) = 5, F'(1) = 5, H(1) = 3, H'(1) = 5
A. If G(z) = F(z)*H(z), then G'(1) = ?
B. If G(w) = F(w)/H(w), then G'(1) = ?"
Should I use the product rule combined with the power rule? If so, I tried to use them in inverse but I got answers that were incorrect. Also, I am struggling to deal with the unknowns that are w and z.
Any assistance or guidance would be highly appreciated!
It seems to me like you're being thrown a little by functions that use variables other than x. Let's review what function notation means, using a simple example. Let f(x) = x - 5. In this example, what does f(x) really mean? Well, it means that
f is a function that takes in the variable x as an input, does some stuff (in this case subtracts 5), and returns an output. We can see that if we "feed"
f an input of x = 1, we get out f(1) = 1 - 5 = -4. Similarly, if we "feed"
f an input of x = 37, we get out f(37) = 37 - 5 = 32.
Now let's suppose we had a different function g(w) = w
2. What does g(w) really mean? In exactly the same way as before, we know that
g is a function that takes the variable w, does some stuff (here, squares it), and returns an output. We can see that if we "feed"
g an input of w = 2, we get out g(2) = 2
2 = 4. Similarly, if we "feed"
g an input of w = -6, we get out g(-6) = (-6)
2 = 36.
So, at the end of the day, there's absolutely no difference between G(x) and G(w) except that we call the variable something different. The only thing that gives me pause is that they use the same function name for both parts, despite them having different definitions. So, the
G from part (a) is a wholly different function from the
G in part (b). But, otherwise, for this problem, you can proceed in exactly the same way you would if all the functions' variables had been x.
Part (a) gives you the function definition for
G in terms of
F and
H. As a quick example, the value of G(1) would be F(1) * H(1) = 5 * 3 = 15. Because
G is defined as a product of two other functions of the same variable, you'd need to use the product rule (although the power rule isn't needed here), and similarly use the quotient rule on part (b).
You say you attempted these but "got [incorrect] answers." So, what did you get for G'(z), as an expression in terms of F(z), F'(z), H(z), and H'(z)? And what did you get for G'(w), as an expression in terms of F(w), F'(w), H(w), and H'(w)? Then what happened when you plugged in z = -1 and w = -1 respectively? Please be as detailed as you can, as we cannot troubleshoot work we can't see. Thank you.
