Find the Derivative - d/dx

Problem

d()/d(x)√(,sin(√(,x)))

Solution

1. Identify the outer function as the square root, which can be rewritten using a fractional exponent.

d()/d(x)*(sin(x(1/2)))(1/2)

2. Apply the power rule and the chain rule to the outermost function.

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

3. Apply the chain rule to the sine function, where the derivative of sin(u) is cos(u)⋅d(u)/d(x)

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

4. Differentiate the innermost square root function using the power rule.

1/2*(sin(x(1/2)))(−1/2)⋅cos(x(1/2))⋅1/2*x(−1/2)

5. Simplify the expression by combining the constants and moving terms with negative exponents to the denominator.

cos(√(,x))/(4√(,x)√(,sin(√(,x))))

Final Answer

d(√(,sin(√(,x))))/d(x)=cos(√(,x))/(4√(,x)√(,sin(√(,x))))

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