LOGARITHMIC DIFFERENTIATION EXAMPLES

Logarithmic differentiation is a method of finding derivatives of some complicated functions, using the properties of logarithms.

There are cases in which differentiating the logarithm of a given function is easier than differentiating the function as it is.

Steps to be followed to find derivative using logarithm. 

Step 1 : 

Take logarithm on both sides of the given equation. 

Step 2 : 

Use the properties of logarithm. 

Step 3 : 

Find derivative with respect to x and solve for dy/dx. 

Find derivative of each of the following with respect to x.

Example 1 :

y = xcosx

Solution :

y = xcosx

Take logarithm on both sides. 

logy = logxcosx

logy = cosxlogx

Find derivative with respect to x. 

(1/y)(dy/dx) = cosx(1/x) + logx(-sinx)

dy/dx = y[(cosx/x) - sinxlogx]

dy/dx = xcosx[(cosx/x) - sinxlogx]

Example 2 :

y = xlogx + (log x)x

Solution :

y = xlogx + (log x)x

Let a = xlogx and b = (log x)x

a = xlogx

loga = log(xlogx)

loga = logxlog x

loga = (log x)----(1)

b = (logx)x

logb = log[(logx)x]

logb = xlog(logx) ----(2)

In (1), find derivative with respect to x.

(1/a)(da/dx) = 2logx(1/x)

 da/dx = 2alogx/x

In (1), find derivative with respect to x.

(1/b)(db/dx) = x(1/logx)(1/x) + log(logx)(1)

db/dx = b[(1/logx) + log(logx)]

y = xlogx + (log x)x

y = a + b

dy/dx = da/dx + db/dx

dy/d = 2alogx/x + b[(1/logx) + log(logx)]

Substitute a = xlogx and b = (logx)x.

dy/dx = (2xlogxlogx)/x] + (logx)x[logx + log(logx)]

Example 3 :

√(xy) = ex - y

Solution :

√(xy) = ex - y

√xy = ex - y

Take logarithm on both sides.

log(√xy) = logex - y

log√x + logy = (x - y)loge

logx1/2 + logy1/2 = x - y

(1/2)logx + (1/2)logy = x - y

Find derivative with respect to x.

(1/2)(1/x) + (1/2)(1/y)(dy/dx) = 1 - dy/dx

1/2x + (1/2y)(dy/dx) = 1 - dy/dx

(1/2y)(dy/dx) + dy/dx = 1 - 1/2x

(dy/dx)(1/2y + 1) = (2x - 1)/2x

(dy/dx)(1 + 2y)/2y = (2x - 1)/2x

dy/dx = (y/x)[(2x - 1)/(1 + 2y)]

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