Question

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

Answer 1

To solve the question given above, you need to know about logarithmic derivation. The logarithmic derivative of the initial function $y = f\left( x \right)$ is the derivative of the logarithmic function. The derivatives of power-exponential functions, or functions of the form, can be computed effectively using this differentiation method. Where $u\left( x \right)$ and $v\left( x \right)$ are differentiable functions of $x$, $y = u\left( x \right)v\left( x \right)$.

Formula used:
We will be using two approaches to solve the question mentioned above. The formulas required to solve the question using two different approaches are:
We can solve this question using the properties of logarithm. We know that $\log {x^n} = n\log x$.
In the second approach we will use the chain rule where: $\dfrac{d}{{dx}}\left( {\log \left( {f\left( x \right)} \right)} \right) = \dfrac{1}{{f\left( x \right)}} \times f'\left( x \right)$

Complete step-by-step answer:
We are given: $y = \log {x^2}$
We will differentiate this function by using the laws of logarithm.
Now, using the first formula: $\log {x^n} = n\log x$
We get: $y = 2\log x$
On differentiating both sides with respect to x, we get:
$\dfrac{{dy}}{{dx}} = 2 \times \dfrac{1}{x}$
$= \dfrac{2}{x}$.
So, our final answer is $\dfrac{2}{x}$
Additional information:
This question can also be solved using the chain rule:
We know that: $\dfrac{d}{{dx}}\left( {\log \left( {f\left( x \right)} \right)} \right) = \dfrac{1}{{f\left( x \right)}} \times f'\left( x \right)$
On differentiating $y = \log {x^2}$ using the chain rule, we get:
$\dfrac{{dy}}{{dx}} = \dfrac{1}{{{x^2}}} \times \dfrac{d}{{dx}}\left( {{x^2}} \right)$
$$
= \dfrac{1}{{{x^2}}} \times 2x \\
= \dfrac{2}{x} \\

The final answer is $$\dfrac{2}{x}$$. **Note:** While solving questions similar to the one given above, remember the chain rule is a method for determining the derivative of composite functions, with the number of functions in the composition determining the number of differentiation steps required. Consider an example: $$f\left( x \right) = \left( {goh} \right)\left( x \right) = g\left( {h\left( x \right)} \right)$$then this becomes $$f'\left( x \right) = g'\left( {h\left( x \right)} \right) \times h'\left( x \right)$$. Since the composite function f is made up of two functions, g and h, you must differentiate $$f\left( x \right)$$ using the derivatives $$g'$$ and $$h'$$.