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

Problem

d()/d(r)r/√(,r2+1)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, apply the quotient rule: d()/d(r)u/v=(vd(u)/d(r)−ud(v)/d(r))/(v2)

2. Assign the variables for the quotient rule where u=r and v=√(,r2+1)=(r2+1)(1/2)

3. Differentiate the numerator u with respect to r

d(r)/d(r)=1

4. Differentiate the denominator v using the chain rule.

d(r2+1)/d(r)=1/2*(r2+1)(−1/2)⋅2*r

d(√(,r2+1))/d(r)=r/√(,r2+1)

5. Substitute these derivatives into the quotient rule formula.

d()/d(r)r/√(,r2+1)=(√(,r2+1)*(1)−r(r/√(,r2+1)))/((√(,r2+1))2)

6. Simplify the numerator by finding a common denominator of √(,r2+1)

d()/d(r)r/√(,r2+1)=(r2+1−r2)/√(,r2+1)/(r2+1)

7. Combine the terms to reach the final simplified form.

d()/d(r)r/√(,r2+1)=1/((r2+1)√(,r2+1))

d()/d(r)r/√(,r2+1)=1/((r2+1)(3/2))

Final Answer

d()/d(r)r/√(,r2+1)=1/((r2+1)(3/2))

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