Higher Differentiation

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Higher Differentiation by Mind Map: Higher Differentiation

1. solve optimisation problems

1.1. show that formula=...

1.1.1. Write the formula you are looking for in terms of y and x

1.1.2. use second piece of information to get y in terms of x

1.1.3. substitute expression for y in terms of x, to get formula in terms of x only

1.2. differentiate function

1.2.1. get into form to differentiate

1.2.1.1. expand any brackets

1.2.1.2. pull apart fractions

1.2.1.2.1. (a+b)/c=a/c+b/c

1.2.1.3. rewrite each term as power of x, using laws of indices

1.2.1.3.1. need to know laws of indices!!

1.2.2. "multiply by power and reduce power by 1"

1.2.3. tidy up

1.3. solve derivative=0

1.3.1. Write down SP => f'(x)=0

1.3.2. set up equation by setting f'(x)=0

1.3.3. if quadratic, solve quadratic

1.3.3.1. factorise

1.3.3.2. solve each factor=0

1.3.4. if linear, solve linear

1.3.4.1. add or subtract same from both sides

1.3.4.2. get to kx=c

1.3.4.3. divide both sides by k to get x=

1.4. find y values for each stationary-point x value

1.4.1. substitute each x value back into f(x)

1.4.1.1. replace each x in the expression for f(x) with the value of x, then do the resulting arithmetic

1.5. Use 2nd derivative test to establish nature of SPs

1.5.1. differentiate f'(x) to get f''(x)

1.5.2. substitute in the x values from the SPs

1.5.3. f''(x)>0 means it is a min TP

1.5.4. f''(x)<0 means it is a max TP

1.5.5. if f''(x)=0, need to use a nature table

1.5.5.1. Do the nature table!!!

1.6. Pull together solution to problem

1.6.1. Read question to see exactly what has been asked

1.6.2. Write down clear final answer in terms of the problem set.