Question

Using two independent methods, find the derivative of ln(1/2x)

Answer 1

we have to find the derivative of ln(1/2x) using two independent methods

Step 1)

we will find the derivative of ln(1/2x) using chain rule

Let assume that y = ln(1/2x)

we can say that,

using chain rule we can say that,

Hence we can say that derivative of ln(1/2x) = -1/x

Step 2)

we will find the derivative of ln(1/2x) using implicit differentiation

Let assume that y = ln(1/2x) and take anti log on both the side we can say that,

we know that e^ln(a) = a hence we can say that,

---------------------------------------------------------1)

Differentiate both the side with respect to x we can say that,

now using implicit differentiation on left hand side we can say that,

from equation 1) we have e^y = 1/2x hence we can say that,

Hence we can say that derivative of ln(1/2x) = -1/x