# What is a derivative

## velocity and tangent question

### velocity question

Average velocity:

Instantaneous velocity:

secant:

tangent:

Or as:

Or as:

Or as:

Or as:

## Ordinary derivative

1.Derivative of :

2.Derivative of :

3.Derivative of :

and in a similar way:

4.Derivative of :

specifically：.

5.Derivative of

specifically：.

## Differentiable and continuous

Derivative must be continuous, continuous may not be derivative.

# Derivative method

## sum, difference, product, quotient method

### Trig derivative

Derivative of

in a similar way:

## Derivative method of inverse function

If the function is monotone and differentiable in the interval ,and ,then differentiable in interval ,and:

## Derivative rule of complex functions

If can be derivable at point ,and can be derivable at point ,then can be derivable at point .The derivative is:

Case 1:If ,for :

Case 2:If ,for .

# higher derivative

## higher derivative

(1)

for :

(2)

### Higher derivatives of sine and cosine

for :

in a similar way,for :

### *Higher derivative of the product function

we know:;

then:;

then:;

so:

This is Leibniz formula.

# Derivative of an implicit function

## General method for differentiating implicit functions

case 1:for function ,for :

(1)derivative for the left-hand side with respect to :

(2)derivative for the right-hand side with respect to :

(3)When we differentiate the left and the right sides of this equation for , we get the same thing:

(4)then:

case 2:for ,please get the equation of the tangent line at point

(1)according to geometric meaning of the derivative,we know the slope of this tangent line:

(2)take the derivative of both sides of this equation with respect to , we get the same thing:

(3)then:

(4)when ,:

(5)The tangent equation is:

(6)Reduction to:

## Take the derivative of an implicit function using logarithmic differentiation

case 1:get derivative of .

(1)Take the logarithm of both sides of this equation:

(2)Take the derivative of both sides of this equation with respect to

(3)then:

## Derivative of parametric equation

The derivative of an equation whose parameters can be eliminated

Derivative of parabolic parametric equation:

(1)Elimination parameters :

(2)Using the general implicit differentiation method,Is omitted.

The derivative of an equation that doesn’t cancel out the parameters

for:

we can :