LOGARITHMIC DIFFERENTIATION EXAMPLES

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

Example 1 :

y = x^cosx

Solution :

y = x^cosx

Take logarithm on both sides. 

logy = logx^cosx

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 = x^cosx[(cosx/x) - sinxlogx]

Example 2 :

y = x^logx + (log x)^x

Solution :

y = x^logx + (log x)^x

Let a = x^logx and b = (log x)^x

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 = x^logx + (log x)^x

y = a + b

dy/dx = da/dx + db/dx

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

Substitute a = x^logx and b = (logx)^x.

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

Example 3 :

√(xy) = e^{x - y}

Solution :

√(xy) = e^{x - y}

√x√y = e^{x - y}

Take logarithm on both sides.

log(√x√y) = loge^{x - y}

log√x + log√y = (x - y)loge

logx^1/2 + logy^1/2 = 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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