Find the Derivative - d/dx x square root of 9-x

Problem

d()/d(x)*x√(,9−x)

Solution

1. Identify the rule needed for the expression, which is a product of x and √(,9−x) We must use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the product where u=x and v=√(,9−x)=(9−x)(1/2)

3. Differentiate each part individually.

d(x)/d(x)=1

d(9−x)/d(x)=1/2*(9−x)(−1/2)⋅(−1)=−1/(2√(,9−x))

4. Apply the product rule formula.

(d(x)√(,9−x))/d(x)=x*(−1/(2√(,9−x)))+√(,9−x)*(1)

5. Simplify the expression by finding a common denominator.

(d(x)√(,9−x))/d(x)=−x/(2√(,9−x))+(2*(9−x))/(2√(,9−x))

6. Combine the terms in the numerator.

(d(x)√(,9−x))/d(x)=(−x+18−2*x)/(2√(,9−x))

(d(x)√(,9−x))/d(x)=(18−3*x)/(2√(,9−x))

Final Answer

(d(x)√(,9−x))/d(x)=(18−3*x)/(2√(,9−x))

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