Find the Derivative - d/dx (5x^2-9x+13)/(2x+4)

Problem

d()/d(x)(5*x2−9*x+13)/(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=5*x2−9*x+13 and the denominator to v=2*x+4

3. Differentiate the numerator to find d(5*x2−9*x+13)/d(x)=10*x−9

4. Differentiate the denominator to find d(2*x+4)/d(x)=2

5. Substitute these components into the quotient rule formula:

((2*x+4)*(10*x−9)−(5*x2−9*x+13)*(2))/((2*x+4)2)

6. Expand the terms in the numerator:

((20*x2−18*x+40*x−36)−(10*x2−18*x+26))/((2*x+4)2)

7. Simplify the numerator by combining like terms:

(20*x2+22*x−36−10*x2+18*x−26)/((2*x+4)2)

(10*x2+40*x−62)/((2*x+4)2)

8. Factor out a common factor of 2 from the numerator and simplify the denominator if possible:

(2*(5*x2+20*x−31))/((2*(x+2))2)

(2*(5*x2+20*x−31))/(4*(x+2)2)

(5*x2+20*x−31)/(2*(x+2)2)

Final Answer

d()/d(x)(5*x2−9*x+13)/(2*x+4)=(5*x2+20*x−31)/(2*(x+2)2)

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