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# Differentiation and Limits

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1. Differentiate
a. Y = 3x + PI^3
b. Y = 1 / (x-3)^3
c. y = (x^4 - x)^3 (3x + 2)^4
d. Y = (1 + x - x^3)^4

2. Compute the following limits.
a. lim(x??)?[(x-2)/(x^2+2)]
b. lim(x??)?[(3x^5- 6x^4+ 2x-6)/(7x^5- 2x^2+ 10,000)]

3. Use limits to compute f"(3) where f (x) = x^2 - 2x +3.

4. a. What is the average rate of change of f(x) given f(x) = -6/x from [1,2] and [1,4].
b. What is the instantaneous rate of change of f(x) when x = 1.

5. Write the equation of the tangent line to the curve y = x^3 - 2x^2 +5 at x = 2.

##### Solution Summary

Differentiation and limits are discussed in the solution.

##### Solution Preview

Differentiate

y = 3x + π^3
dy/dx=3(1)+0=3
y = 1 / (x-3)^3
y=〖(x-3)〗^(-3)
dy/dx=-3(x-3)^(-4) (1)=(-3)/〖(x-3)〗^4
y = (x^4 - x)^3 (3x + 2)^4
dy/dx=(x^4-x)^3 (4) (3x+2)^3 (3)+(3x+2)^4 (3) (x^4-x)^2 (4x^3-1)
dy/dx=12(x^4-x)^3 (3x+2)^3+3(3x+2)^4 (x^4-x)^2 (4x^3-1)
Note that this can be simplified further
dy/dx=3(x^4-x)^2 (3x+2)^3 [4(x^4-x)+(3x+2)(4x^3-1)]
dy/dx=3(x^4-x)^2 (3x+2)^3 (16x^4+8x^3-7x-2)
y = (1 + x  x^3)^4
dy/dx=〖4(1+x-x^3 )〗^3 (1-3x^2)
Compute the following ...

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