Find the Derivative - d/dx arccos(3x)

Problem

d(arccos(3*x))/d(x)

Solution

1. Identify the outer function as arccos(u) and the inner function as u=3*x

2. Apply the chain rule, which states that d(ƒ(u))/d(x)=d(ƒ)/d(u)⋅d(u)/d(x)

3. Recall the derivative formula for the inverse cosine function: d(arccos(u))/d(u)=−1/√(,1−u^2)

4. Differentiate the inner function u=3*x with respect to x to get d(u)/d(x)=3

5. Substitute the components into the chain rule formula:

d(arccos(3*x))/d(x)=−1/√(,1−(3*x)^2)⋅3

6. Simplify the expression by squaring the term inside the square root and multiplying the constants:

d(arccos(3*x))/d(x)=−3/√(,1−9*x^2)

Final Answer

d(arccos(3*x))/d(x)=−3/√(,1−9*x^2)

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