Learn how to find the derivative of sin(x^3). The derivative is d/dx[sin(x^3)].
Below is the graph of f(x) = sin(x^3) 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^3), we first recognize that the function is a composite of a sine function and a cubic function, x^3. This structure requires the application of both the Trigonometric Rule and the Power Rule.
💡 Why this works: The sine function requires the Trigonometric Rule for differentiation, while x^3 requires the Power Rule. Recognizing this allows us to apply both rules appropriately to get the derivative.
First, apply the Trigonometric Rule to sin(x^3). The derivative of sin(u) with respect to u is cos(u). In this case, u = x^3, so we then differentiate x^3 using the Power Rule, which gives 3x^2. The final derivative is obtained by multiplying the result of the Trigonometric Rule by the derivative of x^3.
💡 Why this works: By applying these rules, we get the derivative of sin(x^3) as cos(x^3) multiplied by 3x^2, which is the result of differentiating the cubic term.
After applying the rules, we arrive at the derivative of sin(x^3) as 3x^2 cos(x^3). This expression is simplified, and no further simplification is necessary.
💡 Why this works: The derivative is correct because we have applied the Trigonometric Rule and the Power Rule appropriately. The result is the rate of change of sin(x^3) with respect to x, which is exactly what we set out to find.
[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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