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

Problem

d()/d(x)*cos(√(,3*x))

Solution

1. Identify the outer and inner functions to apply the chain rule. The outer function is cos(u) and the inner function is u=√(,3*x)

2. Apply the chain rule by differentiating the outer function with respect to the inner function. The derivative of cos(u) is −sin(u)

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

3. Rewrite the square root as a power to prepare for differentiation.

√(,3*x)=(3*x)(1/2)

4. Differentiate the inner function using the power rule and the chain rule again for the 3*x term.

d(3*x)/d(x)=1/2*(3*x)(−1/2)⋅3

5. Simplify the derivative of the inner function.

3/(2√(,3*x))

6. Combine the results of the chain rule steps.

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

Final Answer

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

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