Learn how to find the derivative of cos x / x. The derivative is d/dx[cos x / x].
Below is the graph of f(x) = cos x / 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.
We begin by recognizing that cos x / x is a quotient of two functions. The Quotient Rule is necessary because we are dealing with a fraction where one function is divided by another. In this case, the numerator is cos x, and the denominator is x.
💡 Why this works: The Quotient Rule is specifically used to differentiate functions that are in the form of one function divided by another. Here, cos x is the numerator, and x is the denominator, requiring the use of the Quotient Rule.
Now, apply the Quotient Rule, which states that d/dx[f(x)/g(x)] = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2. For cos x / x, f(x) = cos x, and g(x) = x. Differentiate both parts: the derivative of cos x is -sin x, and the derivative of x is 1.
💡 Why this works: By applying the Quotient Rule, we differentiate the numerator and the denominator separately and combine them according to the rule's formula. This gives us d/dx[cos x / x] = (-x * sin x - cos x) / x^2.
Simplifying the expression from Step 2 gives us the final form of the derivative, d/dx[cos x / x] = (-x * sin x - cos x) / x^2. This expression represents the rate of change of the function cos x / x with respect to x.
💡 Why this works: The simplification ensures that the derivative is in its most straightforward form. By verifying the calculations and checking the steps, we confirm that the Quotient Rule was applied correctly, and the derivative is accurate.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ 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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