Learn how to find the derivative of x^2 sin x. The derivative is d/dx[x^2 sin x].
Below is the graph of f(x) = x^2 sin 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.
To differentiate x^2 sin x, we recognize that it involves both a polynomial term (x^2) and a trigonometric function (sin x). The Power Rule applies directly to the x^2 term, while the trigonometric function requires a standard differentiation rule. Additionally, since the function is a product of these two parts, the product rule is needed.
💡 Why this works: The Power Rule applies to the x^2 term because it is a basic polynomial. The product of the two parts (x^2 and sin x) signals that we need the product rule, which is used to differentiate a product of two functions.
First, apply the Power Rule to differentiate x^2. This gives us 2x. Next, differentiate sin x using the standard trigonometric rule, which gives cos x. The product rule tells us to differentiate each part and then combine the results.
💡 Why this works: The derivative of x^2 is 2x, and the derivative of sin x is cos x. Using the product rule, we multiply x^2 by the derivative of sin x (cos x), then add the derivative of x^2 (2x) multiplied by sin x.
The final derivative is a combination of the two differentiated parts: (2x sin x) + (x^2 cos x). This represents the rate of change of the function x^2 sin x at any point.
💡 Why this works: The simplified form, 2x sin x + x^2 cos x, is correct because we have applied the Power Rule and product rule correctly. This form expresses how x^2 sin x changes with respect to 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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