Question

Differentiate the following from first principles: (a) `y=10x` (b) `y=6-5x+x^2`?

Answer 1

a) `dy/dx = 10`

b) `dy/dx= -5+2x`

Explanation:

By definition of the derivative:

`dy/dx = f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h`

Part (A)

With `y=10x` we have;

`dy/dx = lim_(h rarr 0) ( {10(x+h)} -{10x} ) / h`
`" " = lim_(h rarr 0) ( 10x+10h-10x ) / h`
`" " = lim_(h rarr 0) ( 10h ) / h`
`" " = lim_(h rarr 0) ( 10 )`
`" " = 10`

Part (B)

With `y=6-5x+x^2` we have;

`dy/dx = lim_(h rarr 0) ( {6-5(x+h)+(x+h)^2} -{6-5x+x^2} ) / h`
`" " = lim_(h rarr 0) ( 6-5x-5h+x^2+2hx+h^2 -6+5x-x^2 ) / h`
`" " = lim_(h rarr 0) ( -5h+2hx+h^2 ) / h`
`" " = lim_(h rarr 0) ( -5+2x+h )`
`" " = -5+2x`