Question

How do you differentiate \[y = \ln {x^2}\]?

Answer 1

Hint:
We have given a function which is equal to a logarithmic of \[{x^2}\]. We have to find its derivative. Since the variable in the function is \[x\], we differentiate the function with respect to ‘\[x\]’. Firstly, we take derivatives on both sides; the derivative of \[\log {\text{ of }}x\] is \[\dfrac{1}{x}\], we use this property and differentiate. Then, we take the derivative of \[{x^2}\] which will be in the product, since derivative of \[{x^n}\] is \[n{x^{n - 1}}\], so we use this property. Then, we simplify the result and we get the answer.

Complete step by step solution:
The given function is \[y = \ln {x^2}\], we have to find its derivative.
Differentiating both sides with respect to ‘x’
\[ \Rightarrow \dfrac{d}{{dx}}\left( y \right) = \dfrac{d}{{dx}}\left( {\ln {x^2}} \right)\]
Since the derivative of \[\log x\] is \[\dfrac{1}{x}\].
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{{x^2}}} \cdot \dfrac{d}{{dx}}\left( {{x^2}} \right)\]
Now derivative of \[{x^n}\] is \[n{x^{n - 1}}dx\], so
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{{x^2}}} \cdot 2 \times x \cdot \dfrac{{dx}}{{dx}}\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{2x}}{{{x^2}}}\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{2}{x}\]
So, \[\dfrac{{dy}}{{dx}} = \dfrac{2}{x}\]

Derivative of \[y = \ln {x^2}\] is \[\dfrac{2}{x}\].

Note:
In mathematics, logarithm is the inverse function to exponentiation. That means the logarithm of a given number \[x\] is the exponent to which another fixed number, the base \[b\] must be raised to produce that number \[x\]. In the simplest case, the logarithm counts the number occurrences of the same factor in repeated multiplication, example: Since \[ \Rightarrow 1000 = 10 \times 10 \times 10 = {10^3}\].
The logarithm base \[10\] of \[1000\] is \[3\] or \[{\log _{10}}\left( {1000} \right) = 3\]. The logarithm of \[x\] base \[b\] is denoted as \[{\log _b}\left( x \right)\], or without parenthesis, \[{\log _b}x\] or without explicit base \[\log x\].
Generally, exponentiation allows any positive real number as base to be raised to any real power, always producing positive results. So, \[{\log _b}\left( x \right)\], for any two positive real number \[b\] and \[x\], where \[b\] is not equal to \[1\] is always unique real number \[y\].
Let us have a function \[y = f\left( x \right)\] of variable \[x\]. The derivative of the function is the measure of the rate at which the value \[y\] of the function changes with respect to the variable \[x\].