Find dy/dx

Problem

y=√(,x−9)/√(,x+9)

Solution

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

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

3. Differentiate the numerator terms using the power rule and chain rule.

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

4. Factor out the common term 1/2*(x−9)(−1/2)*(x+9)(−1/2) from the numerator to simplify.

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

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

d(y)/d(x)=(1/2*(x−9)(−1/2)*(x+9)(−1/2)⋅18)/(x+9)

6. Reduce the fraction by multiplying 1/2 and 18 then move negative exponents to the denominator.

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

7. Combine the (x+9) terms using exponent rules.

d(y)/d(x)=9/((x−9)(1/2)*(x+9)(3/2))

Final Answer

d(y)/d(x)=9/(√(,x−9)*(x+9)(3/2))

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