# Differentiation

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• Created by: eleanor
• Created on: 22-04-15 17:13
• 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
• 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
• 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
• 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
• Decay: write down the porportion equation dm/dt  -m
• introduce k so dm/dt= -km
• solve to find x