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

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Answer 1 · 2016-06-05T01:51:42

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

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.