Learn how to find the derivative of cos(x^3). The derivative is d/dx[cos(x^3)].
Below is the graph of f(x) = cos(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.
The function cos(x^3) involves a composition of the cosine function and a cubic power of x. To differentiate this, we need to apply both the Trigonometric Rule for cos(x) and the Power Rule for x^3. The Trigonometric Rule tells us how to differentiate cosine functions, and the Power Rule helps us differentiate the inner cubic function.
💡 Why this works: By recognizing that cos(x^3) involves both a trigonometric function and a power of x, we select the appropriate rules to differentiate it. We apply the Trigonometric Rule to the cosine and the Power Rule to the x^3 term.
First, apply the Trigonometric Rule to cos(x), which gives the derivative of -sin(x). Then, apply the chain rule to differentiate the inner function x^3 using the Power Rule. The Power Rule tells us the derivative of x^3 is 3x^2.
💡 Why this works: The derivative of cos(x^3) is found by combining the derivative of the outer cosine function with the derivative of the inner cubic term. This results in -sin(x^3) multiplied by the derivative of x^3, which is 3x^2.
The derivative of cos(x^3) simplifies to -3x^2 sin(x^3). This expression accounts for the changes in cos(x^3) with respect to x, capturing both the trigonometric and power aspects of the function.
💡 Why this works: By applying the Trigonometric Rule and Power Rule, the derivative of cos(x^3) is -3x^2 sin(x^3). This is the correct form, as it combines the effects of both the outer and inner functions in the composition.
[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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