Learn how to find the derivative of sin(x^2). The derivative is 2x*cos(x²).
Below is the graph of f(x) = sin(x^2) 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 sin(x^2), we recognize that it is a composition of two functions: the sine function and x^2. This structure calls for the Chain Rule, as we must differentiate the outer function while keeping the inner function in mind.
💡 Why this works: The Chain Rule is applied because we have a composition of functions. The outer function is sin(u), where u = x^2, and the inner function is x^2. This requires treating the sine part and the x² part separately.
Now we apply the Trigonometric Rule to differentiate the outer sine function, and the Chain Rule to differentiate the inner x² function. First, differentiate sin(u) to get cos(u), then multiply by the derivative of u = x², which is 2x.
💡 Why this works: Using the Trigonometric Rule, we differentiate sin(x²) to get cos(x²), then apply the Chain Rule to differentiate the inner function x², yielding 2x. The final derivative is 2x*cos(x²).
After applying both rules, the derivative simplifies directly to 2x*cos(x²). This is the final result.
💡 Why this works: The final form 2x*cos(x²) is correct because it reflects both the differentiation of the outer sine function and the inner x² function, as required by the Chain Rule and Trigonometric Rule.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Forgetting to multiply by the inner derivative
✅ Correct Approach:
Remember the chain rule: (f(g(x)))' = f'(g(x)) · g'(x)
❌ Common Mistake:
Confusing the derivatives of sin(x) and cos(x)
✅ Correct Approach:
Remember: d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x)
❌ 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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