Question

What is the derivative of $4x$?

Answer 1

We solve this problem by using the simple derivative formulas.
The power rule of differentiation is given as,
$\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$
We use the formula of derivative of product of some constant with variable as,
$\dfrac{d}{dx}\left( kf\left( x \right) \right)=k\dfrac{d}{dx}\left( f\left( x \right) \right)$
Where, $'k'$ is a constant
By using the above rules we find the derivative of required function.

Complete step by step answer:
We are asked to find the derivative of $4x$
Let us assume that the given function as,
$\Rightarrow P\left( x \right)=4x$
Now, let us differentiate both sides with respect to $'x'$ then we get,
$\Rightarrow \dfrac{d}{dx}\left( P\left( x \right) \right)=\dfrac{d}{dx}\left( 4x \right)$
We know that the formula of derivative of product of some constant with variable as,
$\dfrac{d}{dx}\left( kf\left( x \right) \right)=k\dfrac{d}{dx}\left( f\left( x \right) \right)$
Where, $'k'$ is a constant
By using this formula in the above differentiation then we get,
$\Rightarrow \dfrac{d}{dx}\left( P\left( x \right) \right)=4\dfrac{d}{dx}\left( x \right)$
Here, we can see that the function inside the derivative of RHS is $'x'$
We know that this function can be written as $'x'$ power of ‘1’ that is ${{x}^{1}}$
By using this representation in the above differentiation then we get,
$\Rightarrow \dfrac{d}{dx}\left( P\left( x \right) \right)=4\dfrac{d}{dx}\left( {{x}^{1}} \right)$
We know that the power rule of differentiation is given as,
$\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$
By using this rule in above equation then we get,

& \Rightarrow \dfrac{d}{dx}\left( P\left( x \right) \right)=4\left( 1\times {{x}^{1-1}} \right) \\\ & \Rightarrow \dfrac{d}{dx}\left( P\left( x \right) \right)=4\left( {{x}^{0}} \right) \\\ \end{aligned}$$ We know that anything power ‘0’ gives the value of ‘1’ that is ${{x}^{0}}=1$ By using this result in above equation then we get, $\Rightarrow \dfrac{d}{dx}\left( P\left( x \right) \right)=4\left( 1 \right)=4$ Therefore, we can conclude that the required derivative of given function as ‘4’ that is, $\therefore \dfrac{d}{dx}\left( 4x \right)=4$ **Note:** We have a direct shortcut that is direct and standard result for solving this problem. We have some standard results of differentiation. We know that the differentiation of linear function is given as, $\dfrac{d}{dx}\left( ax+b \right)=a$ By using the above result we get the required differentiation as, $\Rightarrow \dfrac{d}{dx}\left( 4x \right)=4$ Therefore, we can conclude that the required derivative of given function as ‘4’ that is, $\therefore \dfrac{d}{dx}\left( 4x \right)=4$