Find the Derivative - d/dx

Problem

d()/d(x)√(,x−9)/√(,x+9)

Solution

1. Rewrite the expression using exponent notation to prepare for differentiation.

√(,x−9)/√(,x+9)=((x−9)/(x+9))(1/2)

2. Apply the chain rule by differentiating the outer power function and then multiplying by the derivative of the inner quotient.

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

3. Apply the quotient rule to the inner expression u/v where u=x−9 and v=x+9

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

4. Simplify the numerator of the quotient rule result.

(x+9−x+9)/((x+9)2)=18/((x+9)2)

5. Combine the parts of the chain rule and simplify the expression.

1/2*((x+9)/(x−9))(1/2)⋅18/((x+9)2)

6. Reduce the constants and combine the terms with the base (x+9)

9/((x−9)(1/2)*(x+9)(2−1/2))

7. Finalize the exponents in the denominator.

9/((x−9)(1/2)*(x+9)(3/2))

Final Answer

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

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