## Properties of derivative -derivative formulas

### Properties of derivative

Functions: f, g, y, u, v

Argument (independent variable): x

Real constant: k

Angle: α

### 1. Derivative of a function:

The derivative of a function y=f(x) measures the rate of change of y with respect to x. Suppose that at some point x ∈ R, the argument of a continuous real function y=f(x) has an increment Δx . Then the increment of the function is equal to Δy = f(x+Δx)− f(x).

The derivative of a function y=f(x) at the point x is defined as the limit of the ratio

$y‘=f‘\left(x\right)=\frac{dy}{dx}=\frac{df\left(x\right)}{dx}$
$=\underset{x\to 0}{\mathrm{lim}}\frac{\Delta y}{\Delta x}=\underset{x\to 0}{\mathrm{lim}}\frac{f\left(x+\Delta x\right)–f\left(x\right)}{\Delta x}$

From a geometrical point of view, the derivative of a function y=f(x) at the point x is equal to the slope of the tangent line to the curve f(x) drawn through this point:

$\frac{dy}{dx}=\mathrm{tan}\alpha$

### 2. Derivative of the sum of two functions:

The derivative of the sum of two functions is equal to the sum of their derivatives:

$\frac{d\left(u+v\right)}{dx}=\frac{du}{dx}+\frac{dv}{dx}$

### 3. Derivative of the difference of two functions:

The derivative of the difference of two functions is equal to the difference of their derivatives:

$\frac{d\left(u–v\right)}{dx}=\frac{du}{dx}–\frac{dv}{dx}$

### 4. Constant factor:

A constant factor can be taken out of a derivative:

$\frac{d\left(ku\right)}{dx}=k\frac{du}{dx}$

### 5. Derivative of the product of two functions:

$\frac{d\left(u·v\right)}{dx}=\frac{du}{dx}·v+u·\frac{dv}{dx}$

### 6. Derivative of the quotient of two functions:

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{\frac{du}{dx}·v–u·\frac{dv}{dx}}{{v}^{2}}$

### 7. Derivative of a composite function (chain rule)

$y=f\left(g\left(x\right)\right),u=g\left(x\right),\frac{dy}{dx}=\frac{dy}{du}·\frac{du}{dx}$

### 8. Derivative of an inverse function:

$\frac{dy}{dx}=\frac{1}{dx/dy}$

Where, x(y) is the inverse function for y(x)

### 10. Logarithmic derivative:

$\frac{dy}{dx}=f\left(x\right)·\frac{d}{dx}\left[\mathrm{ln}f\left(x\right)\right]$