Question

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

Answer 1

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 $x$.