Please help with this differentiation question?

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1 Answer
May 18, 2018

#dy/dx = 12x-96/x^4#

#Delta y ~= 0.88#

Explanation:

Using the linearity of the derivative:

#dy/dx = d/dx ( 6x^2+32/x^3) = 6d/dx(x^2) +32 d/dx(x^-3)#

The derivative of a monomial is given by:

#d/dx (x^n) = nx^(n-1)#, so:

#dy/dx = 12x-96/x^4#

We can approximate the curve with its tangent. The equation of the tanglent line to the continuous curve #y= y(x)# in the point #(x_0,y(x_0))# is:

#y = y(x_0)+ y'(x_0)(x-x_0)#

so the approximate change in value between #x_0# and #x# is:

#y(x) - y(x_0) = y'(x_0)(x-x_0)#

For #x_0=2#:

#y'(x_0) = 12*2 -96/2^4 = 24 -96/16 = 18#

so the approximate change in value is:

#Delta y ~= 18(2.04-2) = 0.72#