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Question: Find the derivative of \( \log ({e^x}) \) using first principle....

Find the derivative of log⁡(ex)\log ({e^x}) using first principle.

Explanation

Solution

In the question they have clearly mentioned to use the first principle of differentiation which is stated as follows,
Given a function y=f(x)y = f(x) , its derivative or the rate of change of f(x)f(x) with respect to xx is defined as
ddxf(x)=f′(x)=lim⁡h→0f(x+h)−f(x)h\dfrac{d}{{dx}}f(x) = f'(x) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f(x + h) - f(x)}}{h}
Where hh is an infinitesimally small positive number.

Complete step by step answer:
Let f(x)=log⁡(ex)f(x) = \log ({e^x})
Let us differentiate f(x)f(x) with respect to xx by using the formula ddxf(x)=lim⁡h→0f(x+h)−f(x)h\dfrac{d}{{dx}}f(x) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f(x + h) - f(x)}}{h}
For finding f(x+h)f(x + h) we replace xx by x+hx + h in the given function.
⇒ddxlog⁡(ex)=lim⁡h→0log⁡(ex+h)−log⁡(ex)h\Rightarrow \dfrac{d}{{dx}}\log ({e^x}) = \mathop {\lim }\limits_{h \to 0} \dfrac{{\log ({e^{x + h}}) - \log ({e^x})}}{h}
From the properties of logarithmic function we know that logarithm of a number containing power can be written as log⁡(xm)=mlog⁡(x)\log ({x^m}) = m\log (x) .
Applying this property to our function we get, ddxlog⁡(ex)=lim⁡h→0(x+h)log⁡e−xlog⁡eh\dfrac{d}{{dx}}\log ({e^x}) = \mathop {\lim }\limits_{h \to 0} \dfrac{{(x + h)\log e - x\log e}}{h}
If the base of logarithm is not mentioned explicitly it is understood that the base is ee and logarithm of a number which is same as the base is always 11 , that is log⁡bb=1{\log _b}b = 1 which implies that log⁡ee=1{\log _e}e = 1 .
Substituting this in the above equation we get,
ddxlog⁡(ex)=lim⁡h→0x+h−xh\dfrac{d}{{dx}}\log ({e^x}) = \mathop {\lim }\limits_{h \to 0} \dfrac{{x + h - x}}{h}
On cancelling xx with opposite signs we have,
ddxlog⁡(ex)=lim⁡h→0hh=lim⁡h→01=1\dfrac{d}{{dx}}\log ({e^x}) = \mathop {\lim }\limits_{h \to 0} \dfrac{h}{h} = \mathop {\lim }\limits_{h \to 0} 1 = 1 .
Therefore the derivative of log⁡(ex)\log ({e^x}) is 11 .

Note: In the question if they do not mention using first principle, we can use direct method to differentiate the function which is easier than first principle. Also the function here contains logarithmic function and when dealing with logarithmic functions all the logarithmic properties must be clearly known, then solving the problem becomes simple.