Question

Differentiate from first principle:

$\left(v\right)-x$

Answer 1

Given:

$f\left(x\right)=-x$

The derivative of a function $f\left(x\right)$ is defined as:

$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$

Putting $f\left(x\right)$ in above expression, we get:

$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{-(x+h)-(-x)}{h}$

$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{-x-h+x}{h}$

$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{-h}{h}$

$\Rightarrow f^{\prime }\left(x\right)=-1$

Therefore, the derivative of $-x$ is $-1.$