Find the Derivative - d/dx (3x-9)/(2x+8)

Problem

d()/d(x)(3*x−9)/(2*x+8)

Solution

1. Identify the numerator u=3*x−9 and the denominator v=2*x+8 to apply the quotient rule.

2. Differentiate the numerator and denominator separately to find u′ and v′

(d(3)*x−9)/d(x)=3

(d(2)*x+8)/d(x)=2

3. Apply the quotient rule formula, which is (v⋅u′−u⋅v′)/(v2)

((2*x+8)*(3)−(3*x−9)*(2))/((2*x+8)2)

4. Distribute the constants in the numerator.

(6*x+24−(6*x−18))/((2*x+8)2)

5. Simplify the numerator by combining like terms.

(6*x+24−6*x+18)/((2*x+8)2)

42/((2*x+8)2)

6. Factor the denominator to simplify the expression further if possible.

42/((2*(x+4))2)

42/(4*(x+4)2)

7. Reduce the fraction by dividing the numerator and the coefficient of the denominator by 2.

21/(2*(x+4)2)

Final Answer

d()/d(x)(3*x−9)/(2*x+8)=21/(2*(x+4)2)

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