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Derivatives of Exponential and Logarithmic Functions

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Derivatives of Exponential and Logarithmic Functions - Lesson Summary

The derivative of ex  w.r.t. x = d/dx (ex) = ex

The derivative of log x w.r.t. x = d/dx (log x) = 1/x

The derivative of a function that is the power of another function i.e

y = f(x) = [u(x)]v(x)

Taking logarithm to base e, we get

log y = log[u(x)]v(x)

⇒ log y = v(x) log u(x)

Differentiating both the sided w.r.t. x, we get

d/dx(log y) = d/dx[v(x) log u(x)]

⇒ 1/y dy/dx = d/dx[v(x) log u(x)]

⇒ 1/y dy/dx = v(x).1/u(x).u'(x) + v'(x).log u(x)

∴ dy/dx = y[v(x).1/u(x).u'(x) + v'(x).log u(x)]

In this case, the logarithmic differentiation method is applicable only if functions f(x) and u(x) are positive.

Logarithmic differentiation is a process by which a complex function can be differentiated by taking logarithm to base e of the function and then differentiating both the sides.


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