derivative question involving x to the power of something?

Question

hi i would like to find out how to solve this problem

i am asking again as in my previous question a few days ago that i posted, some of the answers were incorrect out of the little responses that i did get.

i would like to find the derivative of y=x^5 times 5^x. what would dy/dx=
Thanks for any help. it is much appreciated

Answer 1

Use the product rule: d/dx (f(x)*g(x) = f(x)*g'(x) + f'(x)*g(x)

Let f(x) = x^5; f'(x) = 5*x^4
then g(x) = 5^x g'(x) = 5^x * ln(5)

Put it together

x^5 * 5^x * ln(5) + 5*x^4 * 5^x

Answer 2

you can do product rule OR logarithmic differentiation

PRODUCT RULE:

LOG DIFF: y = x^5 * 5^x
ln y = ln [x^5 * 5^x]
ln y = 5lnx + x ln5
1/y * y' = 5/x + ln5
y ' = y * [ 5/x + ln5 ]
y' = x^5 * 5^x [ 5/x + ln5]

GUARNTEED CORRECT!
:)

Answer 3

y' = (x^5)'(5^x) + (5^x)'(x^5) = 5(x^4)(5^x) + ln5(5^x)(x^5)

The explanation:
let y = f(x) * g(x)
then y' = f'(x)*g(x) + g'(x)*f*(x)

In your case f(x)=x^5 and g(x)=5^x
f'(x) = 5x^4
g'(x) = ln5*(5^x)

Now plug these values into the equation y' = f'(x)*g(x) + g'(x)*f*(x)