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.
Step 1 :
Take logarithm on both sides of the given equation.
Step 2 :
Use the properties of logarithm.
Step 3 :
Differentiate with respect to x and solve for dy/dx.
Differentiate each of the following with respect to x.
Problem 1 :
x^{y} = y^{x}
Solution :
x^{y} = y^{x}
Taking logarithm on both sides.
logx^{y} = logy^{x}
ylogx = xlogy
Differentiate with respect to x.
y(1/x) + logx(dy/dx) = x(1/y)(dy/dx) + logy(1)
y/x + logx(dy/dx) = (x/y)(dy/dx) + logy
logx(dy/dx) - (x/y)(dy/dx) = logy - y/x
(logx - x/y)(dy/dx) = (xlogy - y)/x
[(ylogx - x)/y](dy/dx) = (xlogy - y)/x
dy/dx = (y/x)[(xlogy - y)/(ylogx - x)]
Problem 2 :
y = (cosx)^{logx}
Solution :
y = (cosx)^{logx}
logy = log[(cosx)^{logx}]
logy = (logx)log(cosx)
Differentiate with respect to x.
(1/y)(dy/dx) = logx(1/cosx)(-sinx) + log(cosx)(1/x)
(1/y)(dy/dx) = -logxtanx + log(cosx)/x
(dy/dx) = y[-logxtanx + log(cosx)/x]
dy/dx = (cosx)^{logx}[-logxtanx + log(cos x)/x]
Problem 3 :
(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1
Solution :
Its an implicit function. Since the function is not a complicated one, we don't have to use logarithm to find derivative.
(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1
Differentiate with respect to x.
(2x/a^{2}) + (2y/b^{2})(dy/dx) = 0
(2y/b^{2})(dy/dx) = -(2x/a^{2})
(dy/dx) = -(b^{2}/a^{2})(x/y)
(dy/dx) = -(b^{2}x/a^{2}y)
Problem 4 :
√(x^{2} + y^{2}) = tan^{-1}(y/x)
Solution :
√(x^{2} + y^{2}) = tan^{-1}(y/x)
Differentiate √(x^{2} + y^{2}) with respect to x.
= [1/2√(x^{2}+ y^{2})][2x + 2y(dy/dx)]
= [x + y(dy/dx)]/√(x^{2}+y^{2}) ----(1)
Differentiate tan^{-1}(y/x) with respect to x.
= 1/[1 + (y/x)^{2}](-y/x^{2}) + (1/x)(dy/dx)
= [x^{2}/(x^{2} + y^{2})][x(dy/dx) - y]/x^{2}
= (x(dy/dx) - y)/(x^{2} + y^{2}) ----(2)
(1) = (2)
[x + y(dy/dx)]/√(x^{2}+y^{2}) = (x(dy/dx) - y)/(x^{2} + y^{2})
√(x^{2 }+ y^{2})[x + y(dy/dx)] = x(dy/dx) - y
x√(x^{2 }+ y^{2}) + y√(x^{2 }+ y^{2})(dy/dx) - x(dy/dx) = -y
(dy/dx)(y√(x^{2 }+ y^{2}) - x) = -y - x√(x^{2 }+ y^{2})
(dy/dx) = (x√(x^{2}+y^{2}) + y)/(x - y√(x^{2}+y^{2}))
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