# Differentiation

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- Created by: eleanor
- Created on: 22-04-15 17:13

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- Differentiation
- Implicit Differentiation
- this is needed when you have to differentiate an expression that contains both x and y
- x terms: differentiate as usual y terms: differentiate with respect to y and then write dy/dx after each term
- differentiate each term
- rearrange to the form dy/dx=
- then bring all the dy/dx terms to 1 side and factorise then divide

- rearrange to the form dy/dx=

- Exponential Functions
- learn the result: y=a* = dy/dx=a*ln(a)
- remember you can use this in the chain and product as well

- to prove it take logs of both sides then use implicit differentiation

- learn the result: y=a* = dy/dx=a*ln(a)
- Differential Equations
- connected rates of changes
- this is where you link 3 different variables together using chain rule
- we want to find how the volume changes with time so dV/dt
- v is not function of t it is a function of r so we use chain rule
- 1. define appropriate letters for the variables
- 2. write down the rate of change you were given in the question as a derivative
- 3. write down the derivative you wan to find and link it to the given one using chain rule
- 4. differentiate and apply chain rule

- 3. write down the derivative you wan to find and link it to the given one using chain rule

- 2. write down the rate of change you were given in the question as a derivative

- Proportion Equations
- these describe growth and decay
- growth: 1. write down the proportion equation
- dx/dt x
- 2, introduce a constant k so dx/dt = kx
- 3. solve the equation through integration
- so x = Ae

- 3. solve the equation through integration

- Decay: write down the porportion equation dm/dt -m
- introduce k so dm/dt= -km
- solve to find x

- introduce k so dm/dt= -km

- connected rates of changes

- Implicit Differentiation

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