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

Problem

d()/d(x)*sin(√(,x))

Solution

1. Identify the outer function as sin(u) and the inner function as u=√(,x) which can be rewritten as x^(1/2)

2. Apply the chain rule, which states that the derivative of ƒ*(g(x)) is ƒ^′*(g(x))⋅g(x)^′

3. Differentiate the outer function sin(u) with respect to u to get cos(u)

4. Differentiate the inner function x^(1/2) using the power rule to get 1/2*x^(−1/2)

5. Combine the results and substitute u=√(,x) back into the expression.

d(y)/d(x)=cos(√(,x))⋅1/2*x^(−1/2)

6. Simplify the expression by moving the negative exponent to the denominator and converting it back to radical form.

d(y)/d(x)=cos(√(,x))/(2√(,x))

Final Answer

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

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