Learn how to find the derivative of 1/cos x. The derivative is d/dx[1/cos x].
Below is the graph of f(x) = 1/cos 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.
Recognize that 1/cos x is a quotient of two functions: 1 (the numerator) and cos x (the denominator). The Quotient Rule applies because we are dealing with the derivative of a ratio of two functions.
💡 Why this works: The Quotient Rule is used when differentiating a function that is the ratio of two other functions. In this case, the numerator is 1 (a constant function) and the denominator is cos x (a trigonometric function). The Quotient Rule provides a structured way to find the derivative of such ratios.
To apply the Quotient Rule, differentiate the numerator (1) and the denominator (cos x). The Quotient Rule formula is: (d/dx[u/v]) = (v * du/dx - u * dv/dx) / v².
💡 Why this works: In our case, u = 1 and v = cos x. The derivative of u (1) is 0, and the derivative of v (cos x) is -sin x. Using the Quotient Rule, we get the derivative of 1/cos x as (cos x * 0 - 1 * (-sin x)) / (cos x)², which simplifies to sin x / cos² x.
Simplify the expression sin x / cos² x to its final form and verify that it matches the expected result.
💡 Why this works: The expression sin x / cos² x is the derivative of 1/cos x. This can also be written as sec x * tan x, since sin x / cos² x is equivalent to the product of sec x and tan x. This confirms that the derivative is correct.
[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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