Find the Derivative - d/dx x square root of 3x+1

Problem

d()/d(x)*x√(,3*x+1)

Solution

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

2. Assign the parts of the product where u=x and v=(3*x+1)(1/2)

3. Differentiate u with respect to x to get d(u)/d(x)=1

4. Differentiate v using the chain rule. The derivative of the outer function is 1/2*(3*x+1)(−1/2) and the derivative of the inner function 3*x+1 is 3 Thus, d(v)/d(x)=3/(2√(,3*x+1))

5. Apply the product rule formula by substituting the components: x⋅3/(2√(,3*x+1))+√(,3*x+1)⋅1

6. Simplify the expression by finding a common denominator, which is 2√(,3*x+1)

7. Combine the terms: (3*x)/(2√(,3*x+1))+(2*(3*x+1))/(2√(,3*x+1))=(3*x+6*x+2)/(2√(,3*x+1))

8. Finalize the numerator by adding like terms to get 9*x+2

Final Answer

(d(x)√(,3*x+1))/d(x)=(9*x+2)/(2√(,3*x+1))

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