Learn how to find the derivative of x cos x. The derivative is d/dx[x cos x].
Below is the graph of f(x) = x 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 cos x is a product of two functions: x and cos x. To differentiate this function, we will apply the Power Rule to the x term and treat cos x as a separate trigonometric function.
💡 Why this works: We apply the Power Rule because x is raised to the first power. Since cos x is a standard trigonometric function, we will differentiate it according to its properties while maintaining the structure of the product.
To differentiate x cos x, we use the product rule. The Power Rule will be applied to the x term, and the derivative of cos x, which is -sin x, is applied to the trigonometric part.
💡 Why this works: Differentiating x gives us 1, and the derivative of cos x is -sin x. This leads us to the derivative d/dx[x cos x] = cos x - x sin x.
After applying the rules, we arrive at the derivative d/dx[x cos x] = cos x - x sin x. This result is simplified and verified as the correct derivative form.
💡 Why this works: By applying both the Power Rule and the derivative of cos x, we ensure that the final expression captures the rate of change of x cos x correctly.
[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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