Find the Derivative - d/dx square root of x^2+2

Problem

d(√(,x2+2))/d(x)

Solution

1. Identify the function as a composition of an outer function u(1/2) and an inner function u=x2+2

2. Apply the chain rule, which states that the derivative is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function.

3. Differentiate the outer function using the power rule.

d(u(1/2))/d(u)=1/2*u(−1/2)

4. Differentiate the inner function x2+2 with respect to x

(d(x2)+2)/d(x)=2*x

5. Combine the results and substitute u=x2+2 back into the expression.

1/2*(x2+2)(−1/2)⋅2*x

6. Simplify the expression by canceling the constant factors and rewriting the negative exponent as a square root in the denominator.

x/√(,x2+2)

Final Answer

d(√(,x2+2))/d(x)=x/√(,x2+2)

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