Question

How do you find the derivative of $y = \ln x_{}^3$?

Answer 1

Use the logarithm law $\ln {a^b} = b\ln a$ and formula $\dfrac{d}{{dx}}(\ln x) = \dfrac{1}{x}$ then apply the formula of differentiation $\dfrac{d}{{dx}}(\ln x) = \dfrac{1}{x}$

Complete step by step answer:
Here $y = \ln x_{}^3$,
Using the logarithm law,
$\ln {a^b} = b\ln a$ in $y = \ln x_{}^3$ , where $a$ is $x$ and $b$ is 3 .
Rewriting the equation$y = \ln x_{}^3$ after applying logarithm law
$\Rightarrow y = 3\ln x$
So, $y = 3\ln x$
Differentiating the above equation with respect to $x$
$\Rightarrow \dfrac{d}{{dx}}3\ln x$
First, applying the differentiation formula $\dfrac{{d(kx)}}{{dx}} = k$ where k is constant .
$\Rightarrow 3\dfrac{d}{{dx}}\ln x$ (3 is constant for $\dfrac{d}{{dx}}3\ln x$ )
Now, applying the formula of differentiation, $\dfrac{d}{{dx}}(\ln x) = \dfrac{1}{x}$
$\Rightarrow 3 \times \dfrac{1}{x}$
$\Rightarrow \dfrac{3}{x}$

Thus, derivative of $y = \ln x_{}^3$ is $\dfrac{3}{x}$.

Additional information:
In the graph of $y = \ln x_{}^3$ the slope will be three times the slope of $\ln x$.

Note: $\dfrac{d}{{dx}}(\ln x) = \dfrac{1}{x}$ can also be written as $\dfrac{d}{{dx}}\log x = \dfrac{1}{x}$.
The method of differentiating functions by first using logarithm laws to deduce and then differentiating is called logarithmic differentiation. We use logarithmic differentiation to make it easier to differentiate the logarithm of a function than to differentiate the function itself.
Whenever you come across logarithm questions, try to deduce them using logarithm law to simplify the log expression before differentiating it.
Logarithmic differentiation is used to transform products into sum and division into subtraction using the chain rule and properties of logarithms. The principle of logarithm differentiation can be used in some parts of the differentiation of differentiable functions, providing that these functions are non-zero.
When a base is not given a logarithm function you can automatically assume it as 10 and start solving.
Don’t assume $y = \ln x_{}^3$ as $y = {(\ln x)^3}$ both are different functions . In this $y = \ln x_{}^3$ x power’s is 3 and in this $y = {(\ln x)^3}$ whole power’s is 3.