When do you use a **Logarithm** when taking a derivative?

1 Answer
Mar 22, 2018

See below.

Explanation:

Take the example of #f(x) = ((2 + x)^5(x - 3)^3)/(x + 1)^4#

Differentiating this using the chain/quotient rules would be long and tedious. Therefore, we can use something called logarithmic differentiation.

We start by taking the natural logarithm of both sides.

#lny = ln(((2 + x)^5(x -3)^3)/(x + 1)^4)#

#lny= ln(2 + x)^5 + ln(x- 3)^3 - ln(x +1)^4#

#lny = 5ln(2 + x) + 3ln(x -3) - 4ln(x +1)#

We now differentiate both sides with respect to #x#.

#1/y(dy/dx) = 5/(2 + x) + 3/(x -3) - 4/(x+ 1)#

#dy/dx = y(5/(2 + x) + 3/(x- 3) - 4/(x +1))#

#dy/dx = ((2 +x)^5(x - 3)^3)/(x + 1)^4(5/(2 + x) + 3/(x -3) -4/(x+ 1))#

This is our answer. We can also use it to simplify trig derivatives. Take the example of #g(x) = sec(x)#. Taking the natural logarithm of both sides, we get

#lny = ln(sec(x))#

#lny = ln(1/cos(x))#

#lny = ln(1) - ln(cos(x))#

Now by implicit differentiation on the left and chain rule on the right we have.

#1/y(dy/dx) = 0 - (-sinx) * 1/cosx#

#1/y(dy/dx) = tanx#

#dy/dx =ytanx#

#dy/dx= secxtanx#

Hopefully this helps!