How do you use the chain Rule to find the derivative of #sqrt(2x^3 - 3x- 4)#?

Question

Charles Anctil · Accepted Answer

#y^' = 3/2 * (2x^2 - 1)/sqrt(2x^3 - 3x - 4)#

You can use the chain rule to find the derivative of this function by using #u = 2x^3 - 3x - 4#.

The chain rule tells you that you can differentiate a function #y# that depends on a variable #u#, which in turn depends on another variable #x#, by

#color(blue)(d/dx(y) = d/(du)(y) * d/dx(u))#

Two additional rules of derivation, the power rule and the sum rule will come into play at certain points in the derivation of #y#.

So, if you use #u = 2x^3 - 3x - 4#, then you have

#y = sqrt(u) = u^(1/2)#

So, the derivative of #y# will look like this

#y^' = d/(du)(y) * d/dx(u)#

#y^' = d/(du)(u^(1/2)) * d/dx(2x^3 - 3x - 4)#

The power rule tells you that you can differentiate a variable #x# raised to a power #a# by

#color(blue)(x^a = a * x^(a-1))#

This means that you have

#d/(du)(u^(1/2)) = 1/2 * u^(1/2 - 1)#

#d/(du)(u^(1/2)) = 1/2 * 1/u^(1/2) = 1/2 * 1/sqrt(u)#

The sum rule tells you that the derivative of a function that can be written as a sum of two (or more) functions is equal to the sum of the derivatives of those functions.

For #color(blue)(y = f(x) + g(x) + h(x) + ...)#

you have

#color(blue)(d/dx(y) = f^'(x) + g^'(x) + h^'(x) + ...)#

This means that you can write

#d/dx(2x^3 - 3x - 4) = d/dx(2x^3) + d/dx(-3x) + d/dx(-4)#

#d/dx(2x^3 - 3x - 4) = 2 * 3 x^(3-2) -3 + 0#

#d/dx(2x^3 - 3x - 4) = 6x^2 - 3#

Your original derivative now becomes

#y^' = 1/2 * 1/sqrt(u) * (6x^2 - 3)#

#y^] = 1/2 * 1/sqrt((2x^3 - 3x - 4)) * 3(2x^2 - 1)#

Finally,

#y^' = color(green)(3/2 * (2x^2 - 1)/sqrt(2x^3 - 3x - 4))#