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

Problem

d(√(,2*x−x2))/d(x)

Solution

1. Rewrite the expression using a fractional exponent to make it easier to differentiate.

√(,2*x−x2)=(2*x−x2)(1/2)

2. Apply the power rule and the chain rule, treating 2*x−x2 as the inner function.

d(2*x−x2)/d(x)=1/2*(2*x−x2)(−1/2)⋅d(2*x−x2)/d(x)

3. Differentiate the inner function 2*x−x2 with respect to x

d(2*x−x2)/d(x)=2−2*x

4. Substitute the derivative of the inner function back into the expression.

1/2*(2*x−x2)(−1/2)⋅(2−2*x)

5. Simplify the expression by distributing the 1/2 and moving the negative exponent to the denominator.

(2−2*x)/(2√(,2*x−x2))

6. Factor out a 2 from the numerator to cancel with the denominator.

(2*(1−x))/(2√(,2*x−x2))

7. Finalize the simplification.

(1−x)/√(,2*x−x2)

Final Answer

d(√(,2*x−x2))/d(x)=(1−x)/√(,2*x−x2)

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