Learn how to find the derivative of x sin x. The derivative is d/dx[x sin x].
Below is the graph of f(x) = x 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.
The function x sin x involves both a polynomial term (x) and a trigonometric term (sin x). To differentiate this function, we will use the Power Rule for x and apply the standard differentiation rules for trigonometric functions like sin x.
💡 Why this works: The Power Rule is applied to the polynomial part of the function (x), while the derivative of sin x will be used for the trigonometric part. This combination of rules leads us to the derivative of x sin x.
We differentiate each part of the function separately. The Power Rule gives us the derivative of x, which is 1. The derivative of sin x is cos x. Thus, applying the product rule results in d/dx[x sin x] = sin x + x cos x.
💡 Why this works: By applying the Power Rule to x and the derivative of sin x to the trigonometric portion, we obtain the derivative of x sin x, which is d/dx[x sin x] = sin x + x cos x.
After applying the rules, we end up with the expression sin x + x cos x. This is the derivative of x sin x, which correctly represents how the function changes with respect to x.
💡 Why this works: The final result, sin x + x cos x, is simplified and correctly represents the rate of change of x sin x. Verification can be done by applying the rules step-by-step and checking with derivative calculators.
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❌ 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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