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.
We have the product of two functions, x^2 and cos x, which requires the use of the product rule. Since x^2 is a power of x, the Power Rule applies to this part of the function. The function cos x will be differentiated using its standard derivative rules.
💡 Why this works: The product rule is necessary because the function is the product of x^2 and cos x. The Power Rule applies to x^2 to find its derivative.
To differentiate x^2 cos x, we apply the product rule: d/dx[u(x) v(x)] = u'(x) v(x) + u(x) v'(x). Here, u(x) = x^2 and v(x) = cos x. The derivative of x^2 is 2x, and the derivative of cos x is -sin x.
💡 Why this works: We now differentiate each part: u'(x) = 2x and v'(x) = -sin x. Then, using the product rule, we get the derivative of x^2 cos x: 2x cos x - x^2 sin x.
The derivative of x^2 cos x is 2x cos x - x^2 sin x. This is the final simplified form, where both parts of the product have been differentiated and combined correctly.
💡 Why this works: By following the differentiation rules step-by-step, we obtain the derivative d/dx[x^2 cos x] = 2x cos x - x^2 sin x. This expression is the correct form of the derivative.
[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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