# What is the derivative of -5x?

Aug 14, 2017

$- 5$

#### Explanation:

now the power rule for differentiation is:

$\frac{d}{\mathrm{dx}} \left(a {x}^{n}\right) = a n {x}^{n - 1}$

$\therefore \frac{d}{\mathrm{dx}} \left(- 5 x\right)$

$= \frac{d}{\mathrm{dx}} \left(- 5 {x}^{1}\right)$

$= - 5 \times 1 \times {x}^{1 - 1}$

using the power rule

$= - 5 {x}^{0} = - 5$

if we use the definition

$\frac{\mathrm{dy}}{\mathrm{dx}} = L i {m}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

we have

$\frac{\mathrm{dy}}{\mathrm{dx}} = L i {m}_{h \rightarrow 0} \frac{- 5 \left(x + h\right) - - 5 x}{h}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = L i {m}_{h \rightarrow 0} \frac{- 5 x - 5 h + 5 x}{h}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = L i {m}_{h \rightarrow 0} \frac{- 5 h}{h}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = L i {m}_{h \rightarrow 0} \left(- 5\right) = - 5$

as before

Aug 14, 2017

-5

#### Explanation:

We can say
$f \left(x\right) = - 5 x$
The derivative of $f \left(x\right)$ is defined as

${\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

So,

$\text{The Derivative of f(x)} = {\lim}_{h \to 0} \frac{- 5 x - 5 h - \left(- 5 x\right)}{h}$

$= {\lim}_{h \to 0} \frac{- 5 x + 5 x - 5 h}{h}$

$= {\lim}_{h \to 0} \frac{- 5 h}{h}$

$= - 5$

Hope it'd help.