# What is a derivative

## velocity and tangent question

### velocity question

Average velocity: Instantaneous velocity: ### tangent question

secant: tangent: ## derivative definition

### The derivative at some point Or as: Or as: ### derived function 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： .

## Unilateral derivative  ## The geometric meaning of the derivative

### The equation of the tangent line of a curve ### The normal equation of the curve ## Differentiable and continuous

Derivative must be continuous, continuous may not be derivative.

# Derivative method

## sum, difference, product, quotient method ### How to demonstrate (3) ### Promotion of formula (1)(2) ### 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 . ## Summary of common derivatives # higher derivative

## accelerated speed ## second derivative ## higher derivative

(1) ### The higher derivative of the exponential function

for :

(2) ### Higher derivatives of sine and cosine

for : in a similar way,for : ### Higher derivative of a power function ### Higher derivative of the sum difference function ### *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 : 