Learn how to find the derivative of 1/sin x. The derivative is d/dx[1/sin x].
Below is the graph of f(x) = 1/sin 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.
To differentiate 1/sin x, we first observe that this is a quotient, where 1 is the numerator and sin x is the denominator. Since it is a ratio of two functions, the Quotient Rule applies.
💡 Why this works: The Quotient Rule is used when differentiating a function that can be written as the ratio of two other functions. Here, the numerator is the constant function 1, and the denominator is sin x, which is a function of x.
Now, we apply the Quotient Rule. The Quotient Rule states that if you have a function f(x) = g(x)/h(x), then f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2.
💡 Why this works: For our function 1/sin x, the derivative of the numerator g(x) = 1 is 0, and the derivative of h(x) = sin x is cos x. Applying the Quotient Rule gives us f'(x) = (0 * sin x - 1 * cos x) / (sin x)^2, which simplifies to -cos x / (sin x)^2.
After applying the Quotient Rule, we simplify the expression. The final form of the derivative is f'(x) = -cos x / (sin x)^2.
💡 Why this works: This result is the correct derivative of 1/sin x. It shows how the function changes with respect to x and provides a simplified and accurate expression of its rate of change.
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❌ 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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