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.
The function f(x) = x^2 + sin x is composed of two distinct parts: x^2 and sin x. The Power Rule is applicable to the x^2 term, while the derivative of sin x uses a standard trigonometric rule.
💡 Why this works: We apply the Power Rule to the x^2 part because it is a monomial, and we apply the trigonometric rule to sin x, which results in cos x.
The Power Rule tells us that the derivative of x^2 is 2x. The derivative of sin x is cos x. Combining these gives us the final derivative of x^2 + sin x.
💡 Why this works: By applying the Power Rule to the x^2 term and the derivative rule for sin x, we obtain d/dx[x^2 + sin x] = 2x + cos x.
The derivative d/dx[x^2 + sin x] simplifies to 2x + cos x. This form is the correct representation of the rate of change of the original function.
💡 Why this works: The simplified result is the final derivative. By following the correct differentiation rules, we ensure the solution is mathematically accurate.
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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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