Question

How do you differentiate $y={{(\ln x)}^{4}}$ ?

Answer 1

In the above equation we have to differentiate the given function. Since, in this case the function is a combination of two sub functions therefore, we will use chain rule also known as onion rule. First, we will differentiate the inner function with respect to the given variable after that we will come to the outer function and thus, we get the differentiated result.

Complete step-by-step solution:
The above question belongs to the concept of differentiating logarithmic functions with base ‘e ‘. here we are performing differentiation operation in mathematics or derivatives, it means the sensitivity to change of the function value with respect to a change in its given argument value. There are some rules to be followed while finding derivative of a function. Also, differentiation is a process of finding a function that will give the rate of change of one variable with respect to another.
Now, in the question we have to find the differentiation of $y={{(\ln x)}^{4}}$
To differentiate this function, we have to apply chain rule which means if a given function is in the form of y=f(g(x)) then first we will differentiate g(x) then we will differentiate f(g(x) .
In this question it is given that y=f(g(x)) .
Therefore,
$\dfrac{dy}{dx}=f'\left( g(x) \right)\times g'(x)$

& y={{(\ln x)}^{4}} \\\ & \dfrac{dy}{dx}=f'\left( g(x) \right)\times g'(x) \\\ & \Rightarrow \dfrac{dy}{dx}=4{{(\ln x)}^{3}}\times \dfrac{d}{dx}(\ln x) \\\ & \Rightarrow \dfrac{dy}{dx}=\dfrac{4{{(\ln x)}^{3}}}{x} \\\ \end{aligned}$$ **Hence the differentiation of $$y={{(\ln x)}^{4}}$$ is $$\dfrac{4{{(\ln x)}^{3}}}{x}$$.** **Note:** Differentiate carefully while solving the above question. Keep in mind the basic definition of chain rule and product rule. Remember the concept of differentiations and its formula used in the above question for future use. Avoid silly mistakes and solve step by step.