How do you find the derivative of y=lnsqrt(8x-4)?

2 Answers
May 18, 2017

refer to answers below =)

Explanation:

The derivative of a y= ln(f(x)) function is

dy/dx= color(red)(f'(x))/color(blue)f(x)

The equation is given as y = lnsqrt(8x-4)

Which can be changed to y=ln(8x-4)^(1/2)

The properties of a natural log function allows for its exponent to be "brought" down as such:

y=(1/2)ln(8x-4)

Differentiating it,

dy/dx= (1/2) color(blue)8/color(red)(8x-4) Answer

Note that the numerator is color(red)(f'(x)).

Therefore, differentiating color(blue)(f(x))=8x-4 results in color(red)(f'(x))=8.

Given dy/dx= color(red)(f'(x))/color(blue)f(x),

We will get dy/dx= 8/(8x-4). 1/2

May 18, 2017

dy/dx=4/(8x-4)

Explanation:

"differentiate using the "color(blue)"chain rule"

• d/dx(ln(f(x)))=(f'(x))/(f(x))

• d/dx(f(g(x))=f'(g(x))xxg'(x)larr" for "sqrt(8x-4)

y=lnsqrt(8x-4)=ln(8x-4)^(1/2)

dy/dx=1/((8x-4)^(1/2))xxd/dx(8x-4)^(1/2)

color(white)(dy/dx)=1/((8x-4)^(1/2))xx1/2(8x-4)^(-1/2).d/dx(8x-4)

color(white)(dy/dx)=1/((8x-4)^(1/2))xx4(8x-4)^(-1/2)

color(white)(dy/dx)=4/(8x-4)