Find the Derivative - d/dx (3x^2-9x+11)/(2x+4)

Problem

d()/d(x)(3*x2−9*x+11)/(2*x+4)

Solution

1. Identify the rule needed for the derivative of a quotient, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator to u=3*x2−9*x+11 and the denominator to v=2*x+4

3. Differentiate the numerator to find d(u)/d(x)=6*x−9 and the denominator to find d(v)/d(x)=2

4. Substitute these components into the quotient rule formula:

((2*x+4)*(6*x−9)−(3*x2−9*x+11)*(2))/((2*x+4)2)

5. Expand the terms in the numerator:

((12*x2−18*x+24*x−36)−(6*x2−18*x+22))/((2*x+4)2)

6. Combine like terms in the numerator:

(12*x2+6*x−36−6*x2+18*x−22)/((2*x+4)2)

(6*x2+24*x−58)/((2*x+4)2)

7. Simplify the expression by factoring out a 2 from the numerator and the denominator:

(2*(3*x2+12*x−29))/((2*(x+2))2)

(2*(3*x2+12*x−29))/(4*(x+2)2)

(3*x2+12*x−29)/(2*(x+2)2)

Final Answer

d()/d(x)(3*x2−9*x+11)/(2*x+4)=(3*x2+12*x−29)/(2*(x+2)2)

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