Find dy/dx

Problem

y=√(,x−2)/√(,x+2)

Solution

1. Rewrite the expression using a single radical and fractional exponents to prepare for differentiation.

y=√(,(x−2)/(x+2))

y=((x−2)/(x+2))(1/2)

2. Apply the chain rule by differentiating the outer power function while keeping the inner fraction intact.

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

3. Apply the quotient rule to differentiate the inner fraction u/v where u=x−2 and v=x+2

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

4. Simplify the numerator of the derivative of the inner fraction.

(x+2−x+2)/((x+2)2)=4/((x+2)2)

5. Substitute the result back into the chain rule expression.

d(y)/d(x)=1/2*((x+2)/(x−2))(1/2)⋅4/((x+2)2)

6. Combine the terms and simplify the exponents for the (x+2) terms.

d(y)/d(x)=(2√(,x+2))/(√(,x−2)*(x+2)2)

d(y)/d(x)=2/(√(,x−2)*(x+2)(3/2))

Final Answer

d(y)/d(x)=2/((x−2)(1/2)*(x+2)(3/2))

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