Find the Derivative - d/dx y = square root of 3x+2cos(x)

Problem

d()/d(x)√(,3*x+2*cos(x))

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is the square root, u(1/2) and the inner function is u=3*x+2*cos(x)

2. Apply the Power Rule to the outer function. The derivative of √(,u) is 1/(2√(,u))

3. Differentiate the inner function with respect to x The derivative of 3*x is 3 and the derivative of 2*cos(x) is −2*sin(x)

4. Multiply the derivative of the outer function by the derivative of the inner function according to the Chain Rule.

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

5. Simplify the expression by combining the terms into a single fraction.

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

Final Answer

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

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