Learn how to find the derivative of cos 3x. The derivative is d/dx[cos 3x].
Below is the graph of f(x) = cos 3x and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
The function cos 3x requires the application of the chain rule, which is part of the Power Rule. This is because we have a composition of functions where the outer function is cosine and the inner function is 3x.
💡 Why this works: The chain rule is essentially a form of the Power Rule applied to composite functions. In this case, the derivative of cos(u) with respect to u is -sin(u), and the derivative of 3x with respect to x is 3. Hence, we can apply the chain rule to find the derivative of cos 3x.
To differentiate cos 3x, first, recall that the derivative of cos(u) with respect to u is -sin(u). Then, differentiate the inner function, 3x, which gives 3.
💡 Why this works: By applying the chain rule, we differentiate the outer function, cos 3x, as -sin(3x), and then multiply by the derivative of the inner function, 3. This results in the derivative -3sin 3x.
Simplify the result to get the final derivative of cos 3x, which is -3sin 3x. This is the correct form because we have properly accounted for both the outer and inner functions in the differentiation process.
💡 Why this works: The derivative of cos 3x is -3sin 3x, which is the correct and simplified result. This form directly reflects the changes in the cosine function as x changes, factoring in the multiple of 3 from the inner function.
[Teaching explanation - to be filled]
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❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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