What is the differential of tan(x)?
Understand the Problem
The question is asking for the derivative of the function tan(x), which is a fundamental concept in calculus. We will solve this by applying differentiation rules.
Answer
The derivative of $\tan(x)$ is $\sec^2(x)$.
Answer for screen readers
The derivative of $\tan(x)$ is $\sec^2(x)$.
Steps to Solve
- Identify the function to differentiate
The function we need to differentiate is $\tan(x)$.
- Recall the derivative of the tangent function
The derivative of $\tan(x)$ is a standard result in calculus. It is helpful to remember this result:
$$ \frac{d}{dx}(\tan(x)) = \sec^2(x) $$
- Differentiate the function
Using the result from the previous step, we can directly state that the derivative of $\tan(x)$ with respect to $x$ is:
$$ \frac{d}{dx}(\tan(x)) = \sec^2(x) $$
- Write the final derivative expression
We summarize the result in a clear format:
$$ \frac{d}{dx}(\tan(x)) = \sec^2(x) $$
The derivative of $\tan(x)$ is $\sec^2(x)$.
More Information
The derivative $\sec^2(x)$ plays a significant role in calculus, especially in integration and differential equations. The tangent function is an important trigonometric function, and its derivative is widely used in various applications, including physics and engineering.
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