Logarithmic Differentiation – Examples and Formula

Here you will learn formula of logarithmic differentiation with examples.

Let’s begin –

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 get 

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))

\(\therefore\)  \(dy\over dx\) = 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)

\(\implies\) \(dy\over dx\) = \(x^{sin x}{log x {d\over dx}(sin x) + sin x {d\over dx}(log x)}\)

\(\implies\) \(dy\over dx\) = \(x^{sin x}(cos x.log x + {sin x\over x}\))

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