Let f ( x ) = 3x^6 ln x , find f'(x), find f'(e^2)

Question

Let ⁶

f'(x)= 18x ln⁵ (x)+3x⁵

f'(e²) = _____?

This last part of the question is really confusing me, I know that you would typically enter the value of f'(#) into the equation, but what do I enter whenever it is e² ?

Answer 1

Solution:

y = 3x⁶ lnx , let u = 3x⁶ , and v = lnx then y = uv and y' = f'(x) = u'v+v'u , u' = (6)(3) x^6-1 = 18x⁵

v' = 1/x , y' = ( 18x⁵)(lnx) + (1/x)(3x⁶) = 18x⁵ lnx + 3x⁵

f'(x) = 18x⁵ lnx + 3x⁵ , f'(e²) = 18( e²)⁵ lne² + 3(e²)⁵ = 18 e¹⁰ ln e² + 3 e¹⁰ = 3 e¹⁰ ( 6 ln e² + 1) This can be written as = 3e¹⁰ ( 12 ln e + 1) = 3 e¹⁰ ( 12 (1) + 1) = 3 e¹⁰ ( 13) = 39 e¹⁰