Question

What is the derivative of $-5x$?

Answer 1

To obtain the derivative of the given function we will use Power rule. Firstly we will write the power of the given variable and take the constant outside the derivative sign. Then we will multiply the power with the variables whose power is subtracted by 1. Finally we will simplify it to get the desired answer.

Complete step-by-step answer:
The function given to us is below:
$-5x$
Let us take the function as follows:
$f\left( x \right)=-5x$…..$\left( 1 \right)$
Now we will use Power rule which is given as follows:
$\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$…..$\left( 2 \right)$
Use rule (2) in equation (1) as below:
${f}'\left( x \right)=\dfrac{d}{dx}\left( -5x \right)$
Take the constant term outside the derivative sign and simplify it as follows:
$\begin{aligned} & \Rightarrow {f}'\left( x \right)=-5\dfrac{d}{dx}\left( {{x}^{1}} \right) \\\ & \Rightarrow {f}'\left( x \right)=-5\times 1\times {{x}^{1-1}} \\\ & \Rightarrow {f}'\left( x \right)=-5\times {{x}^{0}} \\\ & \therefore {f}'\left( x \right)=-5 \\\ \end{aligned}$
After simplifying we got the answer as $-5$
Hence derivative of $-5x$ is $-5$

Note: The process of finding the derivative of a function is known as Differentiation. It is used to find the instantaneous rate of change in a function depending on one of its variables. There are different rules for finding derivatives of different function power rules used for a function with expression $x$ raised to any power. Even if the power of the variable is negative we use the same rule. The other rules are Sum rule, Difference rule, product rule, Quotient rule and many more. Another method to find derivatives of such a function we can use is the limit definition of derivatives but it kind of becomes lengthy if the power of the variable is high. Some examples of differentiation is to find motion in a straight line, Acceleration and motion under gravity etc.