Find dy/dx

Problem

y=√(,x−6)/√(,x+6)

Solution

1. Rewrite the expression using fractional exponents to make differentiation easier.

y=((x−6)(1/2))/((x+6)(1/2))

2. Apply the quotient rule which states that d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

d(y)/d(x)=((x+6)(1/2)d(x−6)/d(x)−(x−6)(1/2)d(x+6)/d(x))/(((x+6)(1/2))2)

3. Apply the chain rule to differentiate the numerator and denominator terms.

d(y)/d(x)=((x+6)(1/2)⋅1/2*(x−6)(−1/2)−(x−6)(1/2)⋅1/2*(x+6)(−1/2))/(x+6)

4. Factor out the common term 1/2*(x−6)(−1/2)*(x+6)(−1/2) from the numerator.

d(y)/d(x)=(1/2*(x−6)(−1/2)*(x+6)(−1/2)⋅((x+6)−(x−6)))/(x+6)

5. Simplify the expression inside the parentheses and combine the denominator terms.

d(y)/d(x)=(1/2*(x−6)(−1/2)*(x+6)(−1/2)⋅12)/(x+6)

6. Reduce the fraction by multiplying 1/2 and 12 and moving negative exponents to the denominator.

d(y)/d(x)=6/((x−6)(1/2)*(x+6)(1/2)*(x+6)1)

7. Combine the (x+6) terms using exponent rules.

d(y)/d(x)=6/((x−6)(1/2)*(x+6)(3/2))

Final Answer

d(y)/d(x)=6/(√(,x−6)*(x+6)(3/2))

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