Learn how to find the derivative of cos(x^2). The derivative is d/dx[cos(x^2)].
Below is the graph of f(x) = cos(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.
We need to recognize that cos(x^2) involves a composition of functions, where cos(u) has u = x^2. The Trigonometric Rule applies to the outer cosine function, while the Power Rule will apply to the inner x^2.
💡 Why this works: The Trigonometric Rule tells us that the derivative of cos(u) is -sin(u) multiplied by the derivative of u. The Power Rule applies to x^2, yielding 2x.
Apply the Trigonometric Rule to cos(x^2) and the Power Rule to x^2. First, differentiate cos(u) with respect to u, then multiply by the derivative of x^2.
💡 Why this works: Using the Trigonometric Rule, d/dx[cos(x^2)] becomes -sin(x^2). Then, applying the Power Rule to x^2, we get an additional factor of 2x, resulting in the final derivative of -2x * sin(x^2).
After applying the rules, the derivative is -2x * sin(x^2). This expression tells us how the function cos(x^2) changes as x changes.
💡 Why this works: The derivative d/dx[cos(x^2)] is correct because all the rules have been applied correctly, and no further simplification is needed. The negative sign arises from the derivative of the cosine function.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ 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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