Find the Derivative - d/dr y=r/( square root of r^2+1)

Problem

d()/d(r)r/√(,r^2+1)

Solution

1. Identify the function as a quotient of two functions, u=r and v=√(,r^2+1) which requires the quotient rule d()/d(r)u/v=(vd(u)/d(r)−ud(v)/d(r))/(v^2)

2. Differentiate the numerator u=r with respect to r to get d(u)/d(r)=1

3. Differentiate the denominator v=(r^2+1)^(1/2) using the chain rule to get d(v)/d(r)=1/2*(r^2+1)^(−1/2)⋅2*r which simplifies to d(v)/d(r)=r/√(,r^2+1)

4. Substitute these components into the quotient rule formula:

d(y)/d(r)=(√(,r^2+1)*(1)−r(r/√(,r^2+1)))/((√(,r^2+1))^2)

5. Simplify the numerator by finding a common denominator of √(,r^2+1)

d(y)/d(r)=(r^2+1−r^2)/√(,r^2+1)/(r^2+1)

6. Combine the terms to reach the final simplified form:

d(y)/d(r)=1/((r^2+1)√(,r^2+1))

Final Answer

d()/d(r)r/√(,r^2+1)=1/((r^2+1)^(3/2))

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