Find the Derivative - d/dx y=(x^3+8)/(x+2)

Problem

d()/d(x)(x3+8)/(x+2)

Solution

1. Identify the expression as a rational function where the numerator is a sum of cubes.

2. Factor the numerator using the sum of cubes formula a3+b3=(a+b)*(a2−a*b+b2) where a=x and b=2

x3+8=(x+2)*(x2−2*x+4)

3. Simplify the original function by canceling the common factor (x+2) in the numerator and denominator, assuming x≠−2

y=((x+2)*(x2−2*x+4))/(x+2)

y=x2−2*x+4

4. Differentiate the simplified polynomial expression using the power rule.

d(y)/d(x)=d(x2)/d(x)−(d(2)*x)/d(x)+d(4)/d(x)

5. Calculate the derivative of each term.

d(y)/d(x)=2*x−2

Final Answer

d()/d(x)(x3+8)/(x+2)=2*x−2

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