Find the Derivative - d/dx square root of x+3/( square root of x)

Problem

d()/d(x)*(√(,x)+3/√(,x))

Solution

1. Rewrite the expression using power notation to make differentiation easier.

√(,x)+3/√(,x)=x(1/2)+3*x(−1/2)

2. Apply the sum rule for derivatives, which allows for differentiating each term independently.

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

3. Apply the power rule, d(xn)/d(x)=n*x(n−1) to the first term.

d(x(1/2))/d(x)=1/2*x(−1/2)

4. Apply the power rule and the constant multiple rule to the second term.

(d(3)*x(−1/2))/d(x)=3⋅(−1/2)*x(−3/2)

5. Simplify the coefficients and rewrite the expression using radical notation.

1/(2√(,x))−3/(2*x(3/2))

Final Answer

d()/d(x)*(√(,x)+3/√(,x))=1/(2√(,x))−3/(2*x√(,x))

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