Question

Find the differentiation of the given function: $ y = \log {x^2} $

Answer 1

Hint: We have to find differentiation of a logarithmic function here in the question. Function is given in a standard form: $ y = f(x) $ . So we differentiate $ y $ with respect to $ x $ on both sides of the equation using standard formulas and certain rules for more than one function.

Complete step by step answer:
Firstly, we write the function which is given in the question:
 $ y = \log {x^2} $
We can see that it is given in the standard form of
 $ y = f(x) $
So we differentiate the function of $ y $ with respect to $ x $
 $ \Rightarrow \dfrac{d}{{dx}}(y) = \dfrac{d}{{dx}}(\log {x^2}) $ ---- Equation (1)
In the RHS as we can see we have to differentiate a function of the form $ \log (f(x)) $ , so we apply chain rule when two functions of a variable is given in such a form i.e.
\[\dfrac{d}{{dx}}(\log (f(x))) = \;\dfrac{1}{{f(x)}} \times \dfrac{d}{{dx}}(f(x))\] ----Equation (2)
Where, the differentiation is done in two parts using two formulae i.e.
 $ \dfrac{d}{{dx}}(\log x) = \dfrac{1}{x} $ and
 $ \dfrac{d}{{dx}}(f(x)) = \;\dfrac{{d(f(x))}}{{dx}} $
Now applying the formula of equation (2) in equation (1) we get
 $ \dfrac{{dy}}{{dx}} = \;\dfrac{1}{{{x^2}}} \times \dfrac{d}{{dx}}({x^2}) $ ---- Equation (3)
We know that
 $
  \dfrac{d}{{dx}}({x^n}) = \;n.{x^{n - 1}} \\
   \Rightarrow \dfrac{d}{{dx}}({x^2}) = \;2x \\
  $
Putting the values in equation (3) we get
 $
  \dfrac{{dy}}{{dx}} = \;\dfrac{1}{{{x^2}}} \times 2x \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \;\dfrac{2}{x} \\
  $
This is the result of the differentiation.
Additional information: We can use another approach to solve the question by using the ‘law of logs’.
There are certain properties of a logarithmic function by which we can simplify complex functions into simpler ones.
Here we use the following property of the logarithmic function
 $ \log {x^n} = n\;\log x $
So we use this property to write our function as
 $ \log {x^2} = 2\;\log x $
Now putting this value we differentiate our function like this
 $
  y = \;2\;\log x \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \,2\;\dfrac{d}{{dx}}(\log x) \\
  $
Using the formula
 $ \dfrac{d}{{dx}}(\log x) = \dfrac{1}{x} $ We get
 $
  \dfrac{{dy}}{{dx}} = \;2 \times \dfrac{1}{x} \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{2}{x} \\
  $
This is the result of the differentiation.

Note:
 We think of a composite function as layers of two different functions so while differentiating a composite function we first differentiate the outer layer leaving the inner layer as it is and then we differentiate the inner layer and multiply the two to find the final solution.