Here you will learn formula of logarithmic differentiation with examples.
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Logarithmic Differentiation
We have learnt about the derivatives of the functions of the form \([f(x)]^n\) , \(n^{f(x))}\) and \(n^n\) , where f(x) is a function of x and n is a constant. In this section, we will be mainly discussing derivatives of the functions of the form \([f(x)]^{g(x)}\) where f(x) and g(x) are functions of x x. To find the derivative of this type of functions we proceed as follows :
Let y = \([f(x)]^{g(x)}\). Taking logarithm of both the sides, we getย
log y = g(x) . log{f(x)}
Differrentiating with respect to x, we get
\(1\over y\) \(dy\over dx\) = g(x) \(\times\) \(1\over f(x)\) \(d\over dx\) ((f(x)) + log {f(x)}.\(d\over dx\)(g(x))
\(\therefore\)ย \(dy\over dx\) = y{\({g(x)\over f(x)}\).\(d\over dx\)(f(x)) + log{f(x)}.\(d\over dx\) (g(x))}
Alternatively, we may write
y = \([f(x)]^{g(x)}\) = \(e^{g(x)log{f(x)}}\)
Differentiating with respect to x, we get
\(dy\over dx\) = \(e^{g(x)log{f(x)}}\) { g(x) \(\times\) \(1\over f(x)\) \(d\over dx\) ((f(x)) + log {f(x)}.\(d\over dx\)(g(x)) }
\(\implies\) \(dy\over dx\) = \([f(x)]^{g(x)}\){\({g(x)\over f(x)}\).\(d\over dx\)(f(x)) + log{f(x)}.\(d\over dx\) (g(x))}
Example : Differentiate \(x^x\) with respect to x.
Solution : Let y = \(x^x\). Then,
Taking log both sides,
log y = x.log x
\(\implies\) y = \(e^{x.log x}\)
On differentiating both sides with respect to x, we get
\(dy\over dx\) = \(e^{x.log x}\)\(d\over dx\)(xlogx)
\(\implies\) \(dy\over dx\) = \(x^x{log x \times {d\over dx}(x) + x \times {d\over dx}(log x)}\)
= \(x^x(log x + x\times {1\over x})\)
\(\implies\) \(dy\over dx\) = \(x^x(1 + logx)\)
Example : Differentiate \(x^{sinx}\) with respect to x.
Solution : Let y = \(x^{sinx}\). Then,
Taking log both sides,
log y = sin x.log x
\(\implies\) y = \(e^{sin x.log x}\)
On differentiating both sides with respect to x, we get
\(dy\over dx\) = \(e^{sin x.log x}\)\(d\over dx\)(sin x.log x)
\(\implies\) \(dy\over dx\) = \(x^{sin x}{log x {d\over dx}(sin x) + sin x {d\over dx}(log x)}\)
\(\implies\) \(dy\over dx\) = \(x^{sin x}(cos x.log x + {sin x\over x}\))
