# Higher Differentiation

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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

1.3.3.1. 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!!!