Question

Differentiate the given function. Simplify the answer.

f(x) = 28x*ln√x + 42

f'(x) = 28*ln(x)^(1/2) + (1/(x^(1/2))*(1/2(x)^(-1/2)28x)

f'(x) = 28*ln(x)^(1/2) + (14x*x^(-1/2))/x^(1/2)

f'(x) = 28*ln(x)^(1/2) + (14x^(-1/2))/x^(1/2)

f'(x) = 28*ln(x)^(1/2) + (14x^(-1)

f'(x) = 28*ln(x)^(1/2) + 1/14x

... how do I simplify further to "14(1 + lnx)"??

Answer 1

28 x ln [sqrt (x+42)] ?maybe? I need more parentheses to understand.

[28 x/sqrt(x+42) ](1/2)x/sqrt(x+42) +28 ln(sqrt(x+42)]

14 x^2/(x+42) + 28 ln[sqrt(x+42]

Answer 2

The original equation is...

f(x) = (28x)(ln(sqrt(x)) + 42

Answer 3

Well then, the 42 is irrelevant for our purposes since its derivative is zero.
so
28 x (d/dx ln x^.5) + 28 ln x^.5
28 x [ (1/x^.5) .5 x^-.5 ] + 28 ln x^.5
28 x (.5/x) + 28 ln x^.5
14 + 28 ln sqrt x