Find the Derivative - d/dx (x^2-6x+12)/(x-4)

Problem

d()/d(x)(x2−6*x+12)/(x−4)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions. Let u=x2−6*x+12 and v=x−4

3. Differentiate the individual components.

d(u)/d(x)=2*x−6

d(v)/d(x)=1

4. Substitute these derivatives into the quotient rule formula.

((x−4)*(2*x−6)−(x2−6*x+12)*(1))/((x−4)2)

5. Expand the terms in the numerator.

(x−4)*(2*x−6)=2*x2−6*x−8*x+24=2*x2−14*x+24

−(x2−6*x+12)=−x2+6*x−12

6. Combine like terms in the numerator to simplify the expression.

2*x2−x2−14*x+6*x+24−12=x2−8*x+12

Final Answer

d()/d(x)(x2−6*x+12)/(x−4)=(x2−8*x+12)/((x−4)2)

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