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

Problem

d()/d(x)x/√(,x^2+1)

Solution

1. Identify the function as a quotient y=u/v where u=x and v=√(,x^2+1)=(x^2+1)^(1/2)

2. Apply the quotient rule which states d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

3. Differentiate the numerator to find d(x)/d(x)=1

4. Differentiate the denominator using the chain rule to find d(x^2+1)/d(x)=1/2*(x^2+1)^(−1/2)⋅2*x=x/√(,x^2+1)

5. Substitute these derivatives into the quotient rule formula.

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

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

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

7. Finalize the expression by combining the terms in the denominator.

d(y)/d(x)=1/((x^2+1)√(,x^2+1))

d(y)/d(x)=1/((x^2+1)^(3/2))

Final Answer

d()/d(x)x/√(,x^2+1)=1/((x^2+1)^(3/2))

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