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What is the derivative of secant squared x?

Understand the Problem

The question is asking for the derivative of the function secant squared of x, which is a trigonometric function. To solve this, we will use the chain rule and the known derivative of the secant function.

Answer

$$ \frac{dy}{dx} = 2\sec^2(x) \tan(x) $$
Answer for screen readers

The derivative of the function $y = \sec^2(x)$ is:
$$ \frac{dy}{dx} = 2\sec^2(x) \tan(x) $$

Steps to Solve

  1. Identify the Function
    The function we need to differentiate is $y = \sec^2(x)$.

  2. Use the Chain Rule
    The chain rule states that if you have a function of another function, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.
    Here, we can let $u = \sec(x)$, so $y = u^2$.

  3. Differentiate Outer Function
    Now we differentiate the outer function. The derivative of $u^2$ with respect to $u$ is:
    $$ \frac{dy}{du} = 2u $$

  4. Differentiate Inner Function
    Next, we differentiate the inner function $u = \sec(x)$. The derivative of $\sec(x)$ is:
    $$ \frac{du}{dx} = \sec(x) \tan(x) $$

  5. Combine Derivatives
    Now we can apply the chain rule:
    $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot \sec(x) \tan(x) $$

  6. Substitute for u
    Now we substitute $u = \sec(x)$ back into our equation:
    $$ \frac{dy}{dx} = 2\sec(x) \cdot \sec(x) \tan(x) = 2\sec^2(x) \tan(x) $$

The derivative of the function $y = \sec^2(x)$ is:
$$ \frac{dy}{dx} = 2\sec^2(x) \tan(x) $$

More Information

In trigonometry, the secant function relates to the cosine function as $\sec(x) = \frac{1}{\cos(x)}$. Therefore, the derivative we found can also relate to other trigonometric identities involving sine and cosine.

Tips

  • Confusing the derivative of secant with secant squared. Remember that the derivative of $\sec(x)$ is $\sec(x) \tan(x)$, not $\sec^2(x)$.
  • Forgetting to apply the chain rule correctly when differentiating composite functions.
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