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Calculus problems

1. Differentiate 5x-1/x / 3-ln x **in words, 5x minus 1 over x, divided by 3 minus ln x**

2. Differentiate (ln(3x) - e^2x +1)^-2 **at the end, is to the -2 power**

3. Find the price elasticity of demand at a price of $2 if D(x)=6 / 3x-1 **The price elasticity of demand is PE(P)= -p D' (p) / d(P)

4. Maximize the area of a rectangle in which the length is 20 minus the width.

5. Use the definition of a derivative to differentiate 5/2x

Solution Preview

1. Use quotient rule:

f(x) = [(5x -(1/x) ] / [ 3 - Lnx]

f'(x) = { [ 3 - Lnx] . (5 + (1/x^2) - (5x - (1/x) . (-1/x) } / { (3-Lnx)^2}

= {15 + (3/x^2) - 5 Lnx - (Lnx/x^2) - [ -5+ (1/x^2)] } / { (3-lnx)^2}

= {20 + (2/x^2) - 5Lnx - (Ln x /x^2 } / { (3 - Lnx)^2} <-------------Please simplify

2. Use chain rule:
f(x) = (Ln 3x -e^2x + 1)^ -2

f'(x) = {-2 (Ln 3x -e^2x + 1)^ -3 } . { (3/3x) - e^(2x) }

= {-2 [(1/x) - e^(2x) ] } / { (Ln 3x -e^2x + 1)^3 } <----------------You can simplify

3. You are given that:

D(x) = 6/ (3x-1)

Therefore D'(x) = [(3x-1). 0 - 6(3) ] / (3x-1)^2

= -18 / (3x-1)^2
...

Solution Summary

This solution shows you step by step on how to solve calculus problems involving differentiation. In addtion, it shows you detailed steps on how to do differentiation from the first principles.

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