Derivatives of Exponential and Logarithmic Functions
Let’s start with derivatives of the natural log function.
d/dx(lnu) = 1/u du/dx
“u” is, as usual, some function of x.
So, if we take the derivative of: y = ln(x1/2)
We let u =x1/2 then du/dx =1/2x-1/2
So: dy/dx = 1/x1/2*1/2x-1/2 = 1/2x
Find dy/dx for each of these.
1. y = ln(4x2)
2. y = ln(x3) + (lnx)3
3. y = ln(x2 – 5x + 6)
4. y = ln[(x-1)/x2]1/3
Rewrite as: y = 1/3 ln[(x-1)/x2] = 1/3[ln(x-1) – 2 lnx)]
dy/dx = 1/3[1/(x-1) – 2/x] = (2-x)/3x(x-1)