Find the Derivative - d/dx y=x square root of 3-x^2

Problem

d()/d(x)*x√(,3−x2)

Solution

1. Identify the rule needed for the expression, which is a product of x and √(,3−x2) We will use the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Rewrite the square root as a fractional power to prepare for differentiation.

y=x*(3−x2)(1/2)

3. Apply the product rule by setting u=x and v=(3−x2)(1/2)

d(y)/d(x)=xd(3−x2)/d(x)+(3−x2)(1/2)d(x)/d(x)

4. Apply the chain rule to differentiate (3−x2)(1/2)

d(3−x2)/d(x)=1/2*(3−x2)(−1/2)⋅(−2*x)

5. Simplify the derivative of the second term.

d(3−x2)/d(x)=−x*(3−x2)(−1/2)

6. Substitute the components back into the product rule equation.

d(y)/d(x)=x*(−x*(3−x2)(−1/2))+(3−x2)(1/2)*(1)

7. Combine the terms into a single expression.

d(y)/d(x)=(−x2)/√(,3−x2)+√(,3−x2)

8. Find a common denominator to simplify the result.

d(y)/d(x)=(−x2+(3−x2))/√(,3−x2)

9. Combine like terms in the numerator.

d(y)/d(x)=(3−2*x2)/√(,3−x2)

Final Answer

(d(x)√(,3−x2))/d(x)=(3−2*x2)/√(,3−x2)

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