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

Problem

d(sin(√(,1+x2)))/d(x)

Solution

1. Identify the outer function as sin(u) and the inner function as u=√(,1+x2)

2. Apply the chain rule by differentiating the outer function with respect to u which gives cos(u)

3. Differentiate the inner function u=(1+x2)(1/2) using the power rule and another application of the chain rule.

4. Calculate the derivative of the inner part: d(1+x2)/d(x)=1/2*(1+x2)(−1/2)⋅2*x

5. Simplify the inner derivative to x/√(,1+x2)

6. Combine the results by multiplying the derivative of the outer function by the derivative of the inner function.

Final Answer

d(sin(√(,1+x2)))/d(x)=(x*cos(√(,1+x2)))/√(,1+x2)

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