Find the Derivative - d/dy

Problem

d()/d(y)*ln(((2*y+1)5)/√(,y2+1))

Solution

1. Apply logarithm properties to simplify the expression before differentiating. Use the quotient rule ln(a/b)=ln(a)−ln(b)

ln(((2*y+1)5)/√(,y2+1))=ln((2*y+1)5)−ln(√(,y2+1))

2. Apply power properties of logarithms, ln(an)=n*ln(a) noting that √(,y2+1)=(y2+1)(1/2)

5*ln(2*y+1)−1/2*ln(y2+1)

3. Differentiate the first term using the chain rule, where d(ln(u))/d(y)=1/u⋅d(u)/d(y)

(d(5)*ln(2*y+1))/d(y)=5⋅1/(2*y+1)⋅2

(d(5)*ln(2*y+1))/d(y)=10/(2*y+1)

4. Differentiate the second term using the chain rule.

(d(1/2)*ln(y2+1))/d(y)=1/2⋅1/(y2+1)⋅2*y

(d(1/2)*ln(y2+1))/d(y)=y/(y2+1)

5. Combine the results of the derivatives.

10/(2*y+1)−y/(y2+1)

Final Answer

d()/d(y)*ln(((2*y+1)5)/√(,y2+1))=10/(2*y+1)−y/(y2+1)

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