Logarithmic Differentiation:

If function to the power function or function to the power variable or variable to the power function or variable to the power variable ,then apply logarithm.

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

Now this last limit \frac{b^{h} -1}{h}   is exactly the definition of above derivative f'(x) at x = 0, i.e f'(0). Therefore, the derivative becomes, 

f'(x) = bxf'(0) = bx

So, in case of natural exponential functions, f(x) = ex

Note: In general exponential cases, for example, y = bx, where b is a real number. The derivative for this kind of function is

Question: Differentiate f(x) = 4ex – 5x

Answer: 

The derivation of ex will remain ex, the derivative of 5x will become 5xln(5) as explained above.

Therefore, f'(x) = 4ex – 5xln(x)

Question : Find the value of F'(x) at x=0 when f(x) = 7x + 2ex

Answer: 

Differentiating: f'(x) = 7xln(7) + 2ex

at x=0, f'(0) = 70ln(7) + 2e0

= ln(7) + 2

= 3.945

Question: d/dx(xx) = xx(1+ln x)

Question: Find the value of

 if y = 2x{cos x}.

Solution: Given the function y = 2x{cos x}

Taking logarithm of both the sides, we get

log y = log(2x{cos x})

Now, differentiating both the sides w.r.t by using the chain rule we get,