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
-
Identify the Function
The function we need to differentiate is $y = \sec^2(x)$. -
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 $y = u^2$. -
Differentiate Outer Function
Now we differentiate the outer function. The derivative of $u^2$ with respect to $u$ is:
$$ \frac{dy}{du} = 2u $$ -
Differentiate Inner Function
Next, we differentiate the inner function $u = \sec(x)$. The derivative of $\sec(x)$ is:
$$ \frac{du}{dx} = \sec(x) \tan(x) $$ -
Combine Derivatives
Now we can apply the chain rule:
$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot \sec(x) \tan(x) $$ -
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.
