Find the Derivative - d/dx

Problem

d()/d(x)√(,x−4)/√(,x+4)

Solution

1. Identify the function as a quotient of two functions, u=√(,x−4) and v=√(,x+4) which requires the quotient rule.

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

3. Differentiate the numerator u=(x−4)(1/2) using the power rule and chain rule.

d(√(,x−4))/d(x)=1/(2√(,x−4))

4. Differentiate the denominator v=(x+4)(1/2) using the power rule and chain rule.

d(√(,x+4))/d(x)=1/(2√(,x+4))

5. Substitute these derivatives back into the quotient rule formula.

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

6. Simplify the numerator by finding a common denominator, which is 2√(,x−4)√(,x+4)

d(y)/d(x)=((x+4)−(x−4))/(2√(,x−4)√(,x+4))/(x+4)

7. Combine the terms in the numerator and simplify the fraction.

d(y)/d(x)=8/(2*(x+4)√(,x−4)√(,x+4))

8. Reduce the fraction to its simplest form.

d(y)/d(x)=4/((x+4)√(,x2−16))

Final Answer

d()/d(x)√(,x−4)/√(,x+4)=4/((x+4)√(,x2−16))

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