Question

How do you find the derivative of $y={{x}^{6}}$?

Answer 1

First, we will assume that ‘y’ is f(x). Then we will find the derivative of the given function ${{x}^{6}}$ using the formula $\dfrac{d}{dx}{{x}^{m}}=m{{x}^{m-1}}$ . We will mark it as f’(x). Now, we know that f’(x) would be the required solution of the given question as the derivative of f(x) is f’(x).

Complete step by step solution:
We have been given that $y={{x}^{6}}$.
We know that y is a function and the variable is x. So, we can also write it as $f\left( x \right)={{x}^{6}}$.
We know that for a real constant ‘m’ and a function ${{x}^{m}}$ , the derivative of the function can be determined using the formula $\dfrac{d}{dx}{{x}^{m}}=m{{x}^{m-1}}$.
For a function f(x), it’s derivative can be denoted as f’(x).
Using this, we can write that $f'\left( x \right)=\dfrac{d}{dx}f\left( x \right)$ .
Now, for our function, we have
$f'\left( x \right)=\dfrac{d}{dx}{{x}^{6}}$
Here, we have m=6. Applying the formula for the derivative, we get
$\Rightarrow f'(x)=6{{x}^{6-1}}$
Simplifying the power, we get
$\Rightarrow f'(x)=6{{x}^{5}}$
This is the required solution of the given question.

Note:
We have to understand that we have been given a simple function and we have to use the formula $\dfrac{d}{dx}{{x}^{m}}=m{{x}^{m-1}}$ to obtain the derivative. Some students make mistakes by writing the formula as $\dfrac{d}{dx}{{x}^{m}}=m{{x}^{m+1}}$ . This is a common silly mistake and must be avoided. We can also rewrite the function as $x.{{x}^{5}}$ and then apply the product rule to differentiate the function. But, it is not required since it will only add extra steps and complicate the solution.