To calculate the derivative of secx, where x is an angle measure, we use a combination of trigonometric identities and derivative rules. This process involves five distinct steps, each building upon the previous one. Here’s a detailed breakdown:
Step 1: Understand the Function The function we’re dealing with is sec(x), which is the reciprocal of cos(x). In other words, sec(x) = 1/cos(x). It’s crucial to grasp this fundamental relationship as it will guide our derivative calculation.
Step 2: Apply the Reciprocal Rule The reciprocal rule states that the derivative of a function of the form 1/f(x) is given by -f’(x)/[f(x)]^2. In our case, we have sec(x) = 1/cos(x), so we can apply this rule.
Step 3: Find the Derivative of Cos(x) The derivative of cos(x) is -sin(x). This is a fundamental derivative rule that you should remember.
Step 4: Apply the Reciprocal Rule Again Using the reciprocal rule, we can now find the derivative of sec(x). It becomes -(-sin(x))/cos^2(x), which simplifies to sin(x)/cos^2(x).
Step 5: Rewrite Using Trigonometric Identities We can further simplify this expression by rewriting it using the trigonometric identity sec^2(x) = 1 + tan^2(x). This gives us sin(x)/[1 + tan^2(x)], which is the final derivative of sec(x).
So, in summary, the derivative of sec(x) is sin(x)/[1 + tan^2(x)]. This is a complex derivative, but by breaking it down into these five steps, it becomes more manageable. Each step builds upon the previous one, utilizing fundamental derivative rules and trigonometric identities.
Remember, understanding the function and its relationships is key to successfully calculating derivatives. In this case, recognizing the reciprocal nature of sec(x) and cos(x) allowed us to apply the reciprocal rule, which led us to the final answer.
