Find the Derivative - d/dx

Problem

d()/d(x)√(,x−4)/√(,x+4)

Solution

1. Rewrite the expression using exponent notation to make differentiation easier.

√(,x−4)/√(,x+4)=((x−4)/(x+4))(1/2)

2. Apply the chain rule by differentiating the outer power function and then multiplying by the derivative of the inner quotient.

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

3. Apply the quotient rule to find the derivative of 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. Combine the parts of the chain rule expression.

1/2*((x+4)/(x−4))(1/2)⋅8/((x+4)2)

6. Simplify the constants and the powers of (x+4)

4/((x−4)(1/2)*(x+4)(2−1/2))

7. Finalize the exponents in the denominator.

4/((x−4)(1/2)*(x+4)(3/2))

Final Answer

d()/d(x)√(,x−4)/√(,x+4)=4/(√(,x−4)*(x+4)(3/2))

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