Question

Solve: $\dfrac{d}{dx}{{\left( \log x \right)}^{4}}$.

Answer 1

Hint: We have been asked in the problem to find the differential of a logarithmic function. We will apply the chain rule of differentiation in our problem. The chain rule of differentiation can be given as: $\dfrac{d\left[ f\left\{ g\left( x \right) \right\} \right]}{dx}=\dfrac{d\left[ f\left\{ g\left( x \right) \right\} \right]}{d\left[ g\left( x \right) \right]}\times \dfrac{d\left[ g\left( x \right) \right]}{dx}$ . We will also make use of the fact that the differentiation of any constant is equal to zero. We shall proceed in this manner to get our answer.

Complete step by step answer:
The expression given to us in our problem is equal to: ${{\left( \log x \right)}^{4}}$. Now, let us first assign some terms that we are going to use later in solution.
Let the given term on which we need to operate a differential be given by ‘y’ . Here, ‘y’ is given to us as:
$\Rightarrow y={{\left( \log x \right)}^{4}}$
Then, we need to find the differential of ‘y’ with respect to ‘x’. This can be done as follows:
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d\left[ {{\left( \log x \right)}^{4}} \right]}{dx}$

Applying the chain rule of differentiation, our expression can be written as:
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d\left[ {{\left( \log x \right)}^{4}} \right]}{d\left( \log x \right)}\times \dfrac{d\left( \log x \right)}{dx}$
Now, using the formula for differentiating an expression raised to some power:
$\Rightarrow \dfrac{d{{\theta }^{n}}}{d\theta }=n{{\theta }^{n-1}}$
Our expression can be further simplified into:
$\Rightarrow \dfrac{dy}{dx}=4{{\left( \log x \right)}^{3}}\times \dfrac{d\left( \log x \right)}{dx}$
Now, we will use the formula for differentiating a logarithmic expression, that is given by:
$\Rightarrow d\dfrac{\left( \log x \right)}{dx}=\dfrac{1}{x}$
Therefore, our expression can be further simplified into:
$\begin{align}
  & \Rightarrow \dfrac{dy}{dx}=4{{\left( \log x \right)}^{3}}\times \dfrac{1}{x} \\
 & \therefore \dfrac{dy}{dx}=\dfrac{4{{\log }^{3}}x}{x} \\
\end{align}$
Hence, on solving $\dfrac{d}{dx}{{\left( \log x \right)}^{4}}$, we get the final result as $\dfrac{4{{\log }^{3}}x}{x}$.

Note: While applying the chain rule, we should always make sure we are differentiating our fundamental equation with respect to a correct function. Finding out this function is the key to our solution. Also, while applying different formulas in the same expression, we should make sure that the formulas and calculations are correct. And if nothing is mentioned about the base of our “log” term, we will consider the base of be ‘e’. This is a standard assumption.