Find the derivatives of the following functions: (a) f(x) = (2x^4 - x + 1)(x^5 +

Question

Find the derivatives of the following functions: (a) f(x) = (2x^4 - x + 1)(x^5 + 2) (b) g(x) = x/x^2 -1 Find and simplify d/dx[(f compositefunction g)(x)] given f(x) = 2x^2 -3 and g(x) = x^3 + 5. The cost of producing q units of a particular commodity is given by C(q) = q^2 + 2 q + 10 and the number of units produced after t hours of production is given by q(t) = t^2 + 3t. Find the rate at which cost is changing after 2 hours of production. Use implicit differentiation to find dy/dx given x^3 y^2 - 4x^2 = 1.

Explanation / Answer

Using the rule:

if f = u.v

f' = u'.v + v'.u

if f = u/v

f' = (u'.v - v'.u)/v^2

11A.

f(x) = (2x^4 - x+ 1)*(x^5 + 2)

f' = (8*x^3 - 1)*(x^5 + 2) + (2x^4 - x+ 1)*5*x^4

f' = 18*x^8 - 6*x^5 + 5*x^4 + 16*x^3 - 2

11B.

f(x) = x/(x^2 - 1)

f' = (1*(x^2 - 1) - x*(2x - 0))/(x^2 - 1)^2

f' = (-1 - x^2)/(x^4 + 1 - 2*x^2)

f' = -(x^2 + 1)/(x^4 + 1 - 2*x^2)

12.

f(x) = 2*x^2 - 3

fog(x) = f(g(x))

= f(x^3 + 5)

fog(x) = 2*(x^3 + 5)^2 - 3

d(fog(x))/dx = 2*2*(x^3 + 5)*(3*x^2 + 0) - 0

= 12*x^2*(x^3 + 5)

= 12*x^5 + 60*x^2

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