Question

How do you find the derivative of `f(x) = 4/(sqrt(x))` using the limit definition?

Answer 1

`dy/dx=-2x^(-3/2)`

Explanation:

Let `y=4/sqrtx`

Replace `y` with `(y+Deltay)` and `x` with `(x+Deltax)`

`y=4/sqrtx`

`y+Deltay=4/(sqrt(x+Deltax))`

Subtract `y` and its equivalent `4/sqrtx` from both sides of the equation

`y+Deltay-y=4/(sqrt(x+Deltax))-4/sqrtx`

`Deltay=4/(sqrt(x+Deltax))-4/sqrtx`

Combine the fractions using the LCD`=sqrt(x)*sqrt(x+Delta x)`

`Deltay=4/(sqrt(x+Deltax))-4/sqrtx`

`Deltay=(4sqrtx-4sqrt(x+Deltax))/(sqrtxsqrt(x+Deltax))`

Factor out the `4`

`Deltay=(4(sqrtx-sqrt(x+Deltax)))/(sqrtxsqrt(x+Deltax))`

Multiply the numerator and denominator by `(sqrtx+sqrt(x+Deltax))`

`Deltay=(4(sqrtx-sqrt(x+Deltax)))/(sqrtxsqrt(x+Deltax))*((sqrtx+sqrt(x+Deltax)))/((sqrtx+sqrt(x+Deltax)))`

`Deltay=(4((sqrtx)^2-(sqrt(x+Deltax))^2))/((sqrtxsqrt(x+Deltax))*(sqrtx+sqrt(x+Deltax)))`

`Deltay=(4(x-(x+Deltax)))/((sqrtxsqrt(x+Deltax))*(sqrtx+sqrt(x+Deltax)))`

`Deltay=(4(x-x-Deltax))/((sqrtxsqrt(x+Deltax))*(sqrtx+sqrt(x+Deltax)))`

`Deltay=(4(-Deltax))/((sqrtxsqrt(x+Deltax))*(sqrtx+sqrt(x+Deltax)))`

Divide both sides by `Deltax`

`(Deltay)/(Deltax)=(4(-Deltax))/((sqrtxsqrt(x+Deltax))*(sqrtx+sqrt(x+Deltax)))*1/(Deltax)`

`(Deltay)/(Deltax)=(4(-cancel(Deltax)))/((sqrtxsqrt(x+Deltax))*(sqrtx+sqrt(x+Deltax)))*1/cancel(Deltax)`

`(Deltay)/(Deltax)=(4(-1))/((sqrtxsqrt(x+Deltax))*(sqrtx+sqrt(x+Deltax)))`

Take the limit of both sides as `Deltax rarr 0`

`dy/dx=lim_(Deltax rarr 0)(Deltay)/(Deltax)=lim_(Deltax rarr 0)(4(-1))/((sqrtxsqrt(x+Deltax))*(sqrtx+sqrt(x+Deltax)))`

`dy/dx=(4(-1))/((sqrtxsqrt(x+0))*(sqrtx+sqrt(x+0)))`

`dy/dx=(-4)/(sqrtxsqrtx*(sqrtx+sqrtx))`

`dy/dx=(-4)/(x*(2sqrtx))`

`dy/dx=(-2)/x^(3/2)`

`dy/dx=-2x^(-3/2)`

God bless....I hope the explanation is useful.