Given that f(x)=x^5h(x), h(-1)=5, h'(-1)=8, what is f'(-1)?

Question

f(x) = x⁵ h(x)

h(-1) = 5

h'(-1) = 8

Calculate f'(-1).

It tells me to use the product rule, and the power rule but I can't seem to figure it out, and I've gotten multiple people to help me and I keep getting drastically different answers and they are all wrong. I also need to understand this material, so could somebody possibly explain the product rule a little bit when helping with this problem?

Answer 1

Product goes as follows:

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

Which let's start simpler, let's make

g(x) = x

h(x) = x

Let's take their derivative separately,

g'(x) = 1

h'(x) = 1

Okay, now let's see what Product Rule says:

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

The quick saying I was taught was,

"The derivative of a product is equal to the derivative of the first times the second plus the second times the derivative of the first"

It's even more interesting when there's more than just 2 functions multipled together

f(x) = g(x) * h(x) * a(x) * b(x) * r(x) * s(x)

f'(x) = g'(x) * h(x) * a(x) * b(x) * r(x) * s(x)

+ g(x) * h'(x) * a(x) * b(x) * r(x) * s(x)

+ g(x) * h(x) * a'(x) * b(x) * r(x) * s(x)

+ g(x) * h(x) * a(x) * b'(x) * r(x) * s(x)

+ g(x) * h(x) * a(x) * b(x) * r'(x) * s(x)

+ g(x) * h(x) * a(x) * b(x) * r(x) * s'(x)

The derivative of all g, h, a, b, r, and s just goes down the line and you add all them together

Now, let's make it specific to this problem

g(x) = x⁵

h(x) = h(x)

g'(x) = 5 x⁴

h'(x) = h'(x)

Thus,

f'(x) = x⁵ h'(x) + 5 x⁴ h(x)

f'(-1) = (-1)⁵ h'(-1) + 5 (-1)⁴ h(-1)

f'(-1) = - 8 + 5 * 5 = 25 - 8

∴ f'(-1) = 17

I hope this helps! Please message me in the comments if you have any questions, comments, or concerns!

Answer 2

Product Rule: If y = uv then dy/dx = u dv/dx + v du/dx

f(x) = x⁵ h(x)

f'(x) = x⁵ h'(x) +h(x)(5x⁴)

Now subbing in the values given

h(-1) = 5

h'(-1) = 8

f'(x) = (-1)⁵(8) +(5)(5(-1)⁴) Simplify

f'(x) = (-8)+25 = 17