Learn how to find the derivative of sin x / x. The derivative is d/dx[sin x / x].
Below is the graph of f(x) = sin 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.
sin x / x is a ratio of two functions, sin x (the numerator) and x (the denominator), which means we must use the Quotient Rule to differentiate this function.
💡 Why this works: The Quotient Rule is applied when differentiating a function that is the ratio of two other functions. For sin x / x, we identify the numerator as sin x and the denominator as x, setting us up to use the Quotient Rule.
The Quotient Rule states that d/dx[g(x) / h(x)] = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2. For sin x / x, g(x) = sin x and h(x) = x.
💡 Why this works: Differentiating sin x gives cos x, and differentiating x gives 1. Applying the Quotient Rule, we get the derivative of sin x / x as (cos x * x - sin x * 1) / x^2.
After applying the Quotient Rule, simplify the result to (x cos x - sin x) / x^2.
💡 Why this works: The final derivative, (x cos x - sin x) / x^2, is the correct derivative of sin x / x. It represents the rate of change of the function and can be used to analyze its behavior across the domain.
[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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