Learn how to find the derivative of x^2 + cos x. The derivative is d/dx[x^2 + cos x].
Below is the graph of f(x) = x^2 + cos 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 x^2 + cos x is made up of two terms: a polynomial term (x^2) and a trigonometric term (cos x). The Power Rule is used to differentiate the x^2 term because it is a polynomial, and the standard derivative rule for trigonometric functions is applied to cos x.
💡 Why this works: The Power Rule applies to the x^2 term because it follows the form of a polynomial term. The cosine function, being a standard trigonometric function, is differentiated using its known derivative, -sin x.
Differentiate each term separately. The Power Rule gives us the derivative of x^2 as 2x, and the derivative of cos x is -sin x.
💡 Why this works: By applying the Power Rule to x^2, we obtain 2x. Differentiating cos x gives -sin x, so the derivative of the entire function is f'(x) = 2x - sin x.
The derivative is already in its simplest form, 2x - sin x, and no further simplification is needed.
💡 Why this works: By applying the rules correctly, the final derivative f'(x) = 2x - sin x is verified as correct. This derivative fully describes the rate of change of the function x^2 + cos x.
[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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