Find f(x)= lnx / x^12, find f'(1).

ln x / x^12

find f'(1).

To make this easier to differentiate, re-write f(x) as a product with a negative exponent of x to get:
f(x) = ln(x)*x^(-12).

Using the Product Rule:
f'(x) = (1/x)x^(-12) + ln(x)[-12x^(-13)]
= 1/x^13 - 12ln(x)/x^13
= [1 - 12ln(x)]/x^13.

At x = 1:
f'(1) = [1 - 12ln(1)]/1^13
= [1 - 12(0)]/1
= 1.

I hope this helps!

The derivative of ln(x) is 1/x. Knowing that, use the product rule.
f (x) = ln(x) * x⁻¹²
f ' (x) = ln(x) * (-12x⁻¹³) + (x⁻¹²) * (1/x)
= -12ln(x) * x⁻¹³ + x⁻¹³
= x⁻¹³ * (1 - 12ln(x))

f ' (1) = (1)⁻¹³ * (1 - 12ln(1))= 1 * (1 - 0) = 1

f(x) = ln(x)/(x^12)

f '(x) = [x^(12)*(1/x) - 12x^(11)*ln(x)]/(x^24)

f '(x) = (1 - 12*ln(x))/(x^(13))

f '(1) = 1/1 = 1