Find the Derivative - d/dx y=arctan( square root of x)

Problem

d()/d(x)*arctan(√(,x))

Solution

1. Identify the outer and inner functions to apply the Chain Rule. The outer function is arctan(u) and the inner function is u=√(,x)

2. Apply the derivative formula for the arctangent function, which is d(arctan(u))/d(u)=1/(1+u2)

3. Differentiate the inner function u=√(,x)=x(1/2) using the Power Rule, which gives d(u)/d(x)=1/2*x(−1/2)=1/(2√(,x))

4. Combine the results using the Chain Rule formula d(y)/d(x)=d(y)/d(u)⋅d(u)/d(x)

d(y)/d(x)=1/(1+(√(,x))2)⋅1/(2√(,x))

5. Simplify the expression by squaring the square root and multiplying the denominators.

d(y)/d(x)=1/(1+x)⋅1/(2√(,x))

Final Answer

d(arctan(√(,x)))/d(x)=1/(2√(,x)*(1+x))

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