Learn how to find the derivative of sec x. The derivative is d/dx[sec x].
Below is the graph of f(x) = sec 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.
Sec x is a function of the form (cos x)^{-1}, so the Power Rule applies when differentiating it. This structure involves a negative exponent of a trigonometric function, which is why we can apply the Power Rule after transforming the function.
💡 Why this works: We first rewrite sec x as (cos x)^{-1} to prepare it for differentiation. This form fits the structure where the Power Rule can be used, applying the chain rule for composite functions.
To differentiate sec x, we apply the Power Rule to (cos x)^{-1}, and then use the chain rule. The Power Rule tells us to bring the exponent down and reduce it by one, while the chain rule accounts for the derivative of cos x.
💡 Why this works: Differentiating (cos x)^{-1} gives us -1 * (cos x)^{-2} * d/dx[cos x]. Since the derivative of cos x is -sin x, this leads to sec x tan x as the final derivative.
After applying the differentiation rules, we simplify the expression to arrive at d/dx[sec x] = sec x tan x.
💡 Why this works: This final form is the correct derivative of sec x. It is verified by ensuring the application of both the Power Rule and chain rule were correctly followed, confirming that the result accurately represents the rate of change of sec x.
[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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