Question:

Derivative of $ {{\log }_{10}}\,x $ with respect to $ {{x}^{2}} $ is

Updated On: Jun 23, 2024
  • $ 2{{x}^{2}}\,{{\log }_{e}}\,10 $
  • $ \frac{{{\log }_{10}}\,e}{2{{x}^{2}}} $
  • $ \frac{{{\log }_{e}}\,10}{2{{x}^{2}}} $
  • $ {{x}^{2}}\,{{\log }_{e}}\,10 $
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The Correct Option is B

Solution and Explanation

Let $ u={{\log }_{10}}x=\frac{{{\log }_{e}}x}{{{\log }_{e}}10}={{\log }_{10}}\,e\,{{\log }_{e}}x $
$ \therefore $ $ \frac{du}{dx}=\frac{{{\log }_{10}}\,e}{x} $
and $ v={{x}^{2}} $
$ \therefore $ $ \frac{dv}{dx}=2x $
Now, $ \frac{du}{dv}=\frac{{{\log }_{10}}e}{x+2x}=\frac{{{\log }_{10}}e}{2{{x}^{2}}} $
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Concepts Used:

Continuity

A function is said to be continuous at a point x = a,  if

limx→a

f(x) Exists, and

limx→a

f(x) = f(a)

It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.

If the function is undefined or does not exist, then we say that the function is discontinuous.

Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:

  • The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
  • The limit of the function as x approaches a, exists.
  • The limit of the function as x approaches a, must be equal to the function value at x = a.