Find dy/dx

Problem

y=√(,x−4)/√(,x+4)

Solution

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

y=((x−4)/(x+4))(1/2)

2. Apply the chain rule to differentiate the outer power function.

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

3. Apply the quotient rule to the inner fraction u/v where u=x−4 and v=x+4

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

4. Simplify the numerator of the quotient rule result.

(x+4−x+4)/((x+4)2)=8/((x+4)2)

5. Substitute the derivative of the inner function back into the chain rule expression.

d(y)/d(x)=1/2*((x+4)/(x−4))(1/2)⋅8/((x+4)2)

6. Combine the terms and simplify the exponents for the (x+4) terms.

d(y)/d(x)=4/((x−4)(1/2)*(x+4)(2−1/2))

7. Finalize the expression by simplifying the exponent 2 - 1/2 = 3/2$.

d(y)/d(x)=4/(√(,x−4)*(x+4)(3/2))

Final Answer

d(y)/d(x)=4/(√(,x−4)*(x+4)(3/2))

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