Apply laws of logarithms to simplify the function f(x)=ln√9-x^2/4+x^2. then find its derivative.?

Answer 1

f(x) = ln [sqrt(9 - x^2) / (4 + x^2) ]

First things first, use the fact that
log[base b](a/c) = log[base b](a) - log[base b](c)

f(x) = ln [sqrt(9 - x^2)] - ln(4 + x^2)

Now, convert the square root as a power to 1/2.

f(x) = ln [ (9 - x^2)^(1/2) ] - ln(4 + x^2)

As per log property log[base b](a^c) = c * log[base b](a),

f(x) = (1/2) ln (9 - x^2) - ln(4 + x^2)

Now, let's take the derivative. Note that the derivative of ln(x) is 1/x, so that's what we're going to use, along with the chain rule. Also, keep in mind that the (1/2) can be ignored when taking the derivative.

f'(x) = (1/2) [ 1/(9 - x^2) ] (-2x) - [1/(4 + x^2)] (2x)

Simplifying, we have

f'(x) = -x/(9 - x^2) - 2x/(4 + x^2)