Notes On Derivatives of Exponential and Logarithmic Functions - CBSE Class 12 Maths
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.

#### 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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