Find the Derivative - d/dx square root of xsin(x)

Problem

d(√(,x*sin(x)))/d(x)

Solution

1. Identify the outer function as a square root, which can be rewritten as a power of 1/2

√(,x*sin(x))=(x*sin(x))1/2

2. Apply the chain rule by differentiating the outer power function and multiplying by the derivative of the inner function.

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

3. Apply the product rule to find the derivative of the inner function x*sin(x)

(d(x)*sin(x))/d(x)=sin(x)⋅d(x)/d(x)+x⋅d(sin(x))/d(x)

4. Evaluate the derivatives of the individual terms.

(d(x)*sin(x))/d(x)=sin(x)+x*cos(x)

5. Substitute the result of the product rule back into the chain rule expression.

d(√(,x*sin(x)))/d(x)=1/2*(x*sin(x))(−1/2)*(sin(x)+x*cos(x))

.

6. Simplify the expression by moving the negative exponent to the denominator and combining terms.

d(√(,x*sin(x)))/d(x)=(sin(x)+x*cos(x))/(2√(,x*sin(x)))

Final Answer

d(√(,x*sin(x)))/d(x)=(sin(x)+x*cos(x))/(2√(,x*sin(x)))

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