Learn how to find the derivative of sin 3x. The derivative is d/dx[sin 3x].
Below is the graph of f(x) = sin 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 sin 3x is a composite function, where 3x is the argument of the sine function. The Power Rule applies here because we need to differentiate sin 3x as a composition of functions, involving both the sine and the linear function 3x.
💡 Why this works: To apply the Power Rule correctly, we recognize that the sine function is being evaluated at 3x, which involves the chain rule. The Power Rule is used to differentiate the inner function 3x, and the outer function, sin, will be treated accordingly with its standard derivative, cos.
We begin by differentiating the outer sine function and applying the chain rule. The derivative of sin(u) is cos(u), so we differentiate sin 3x to get cos(3x). Then, the derivative of the inner function, 3x, is simply 3.
💡 Why this works: By using the Power Rule and chain rule, we differentiate the composite function sin 3x step by step, ultimately obtaining the derivative of sin 3x as 3 cos 3x.
The result from applying the Power Rule is 3 cos 3x. This is the final derivative of sin 3x. There are no further simplifications necessary.
💡 Why this works: After applying the rules correctly, we arrive at the final form, which represents the rate of change of sin 3x at any point. This derivative is correct because it follows the standard differentiation procedure for trigonometric functions.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ 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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