Learn how to find the derivative of cos^2 x. The derivative is d/dx[cos^2 x].
Below is the graph of f(x) = cos^2 x 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^2 x involves a composition of the cosine function and a power. Since the power is 2, the Power Rule is the appropriate method to differentiate this function.
💡 Why this works: The Power Rule applies because cos^2 x is a function of the form f(x) = [g(x)]^n, where n = 2 and g(x) = cos(x). We will differentiate the outer function first, then the inner function (cos(x)).
The Power Rule is applied by first differentiating the outer function, which is the square of cos(x). This gives us 2 * cos(x) raised to the first power, multiplied by the derivative of cos(x).
💡 Why this works: The derivative of cos^2 x is calculated as d/dx[cos^2 x] = 2 * cos(x) * (-sin(x)), where -sin(x) is the derivative of cos(x). The Power Rule allows us to handle this composition step-by-step.
After applying the rules, we simplify the expression to -2 * cos(x) * sin(x). This is the correct derivative of cos^2 x.
💡 Why this works: The final derivative, d/dx[cos^2 x] = -2 * cos(x) * sin(x), is correct because it follows the steps outlined by the Power Rule and accounts for both the outer and inner 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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