Calculate f'(-1)

f(x) = x^7 * h(x)
h(-1) = 5
h'(-1) = 8

Answer is 27, but I got no idea how to get there.

first, recall chain rule, since there there is a function of x multiplied by another function of x (that is, x^7 and h(x)),, given a function f(x)=g(x)*h(x)

f'(x) = g'(x)*h(x) + g(x)*h'(x)

therefore, to get derivative of f(x)=x^7 *h(x), first get the derivative of x^7 multiplied by h(x) plus the derivative of h(x) [which is h'(x)] multiplied by x^7,,

since it is evaluated at -1, substitute values for h(-1) and h'(-1), which is given in the problem,,

so there,, please ask questions if there's something you did not understand,, :)

find f' (x) first of all using the product rule assuming we are differentiating with respect to x

f' (x) = x^7 (h' (x)) + 7x^6 (h(x))

so f'-1) = (-1)^7 (h'(-1)) + 7(-1)^6 (h(-1))
= -1(5) + 7(-1)^6 (8)
=-5 + 56
= 51

I don't see how they got 27

I see my error, I substituted the wrong way, should have been ...

so f'-1) = (-1)^7 (h'(-1)) + 7(-1)^6 (h(-1))
= -1(8) + 7(-1)^6 (5)
= -8 + 35
= 27

To find the derivative of the function f(x) = x^7 * h(x), we can use the product rule. The product rule states that if we have two differentiable functions, u(x) and v(x), then the derivative of their product is given by:

(f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x)

In this case, let's let u(x) = x^7 and v(x) = h(x). Then, we can rewrite the function as f(x) = u(x) * v(x). Now, we need to find the derivatives of u(x), v(x), and use the product rule to find f'(x).

Derivative of u(x):
u(x) = x^7
To find the derivative of x^n, where n is a constant, we can use the power rule. The power rule states that the derivative of x^n is given by n*x^(n-1).
So, the derivative of u(x) = x^7 is u'(x) = 7*x^(7-1) = 7x^6.

Derivative of v(x):
v(x) = h(x)
We are given that h'(-1) = 8, but we need h'(x). Since h(x) is not explicitly given, we cannot directly find its derivative. However, we are given that h(-1) = 5. With this information, we can use the definition of the derivative:

h'(x) = [h(x + h) - h(x)] / h

Here, we can substitute x = -1 and h = 1 to find h'(-1):

h'(-1) = [h(-1 + 1) - h(-1)] / 1
= [h(0) - h(-1)] / 1
= [h(0) - 5] / 1
= [h(0) - 5]

So, we cannot determine the exact value of h'(-1) without knowing the function h(x) explicitly or additional information about it.

Now, let's use the product rule to find f'(x):
f'(x) = u'(x) * v(x) + u(x) * v'(x)
= 7x^6 * h(x) + x^7 * h'(x)

To calculate f'(-1), we substitute x = -1 and use the information given:
f'(-1) = 7(-1)^6 * h(-1) + (-1)^7 * h'(-1)
= 7(1) * 5 + (-1) * [h(0) - 5]
= 35 + (-1) * [h(0) - 5]

Since we do not have information about h(0) or h'(0), we cannot determine h(0) - 5. Therefore, we cannot find the exact value of f'(-1) based on the information given.