Find the Derivative - d/dx 1-cos(9x)

Problem

d()/d(x)*(1−cos(9*x))

Solution

1. Apply the sum rule for differentiation, which allows for the derivative of each term to be taken separately.

d()/d(x)*(1−cos(9*x))=d(1)/d(x)−d(cos(9*x))/d(x)

2. Differentiate the constant term, noting that the derivative of any constant is zero.

d(1)/d(x)=0

3. Apply the chain rule to the trigonometric term, where the outer function is cos(u) and the inner function is u=9*x

d(cos(9*x))/d(x)=−sin(9*x)⋅(d(9)*x)/d(x)

4. Differentiate the inner function 9*x with respect to x

(d(9)*x)/d(x)=9

5. Combine the results and simplify the signs, noting that the negative from the derivative of cosine and the negative from the original expression cancel out.

0−(−sin(9*x)⋅9)=9*sin(9*x)

Final Answer

d()/d(x)*(1−cos(9*x))=9*sin(9*x)

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