Differentiation of tan x
Understand the Problem
The question is asking for the derivative of the tangent function with respect to x, which involves applying the rules of differentiation in calculus.
Answer
The derivative of the tangent function is $f'(x) = \sec^2(x)$.
Answer for screen readers
The derivative of the tangent function is $f'(x) = \sec^2(x)$.
Steps to Solve
- Identify the function to differentiate
The function we want to differentiate is the tangent function, which is written as $f(x) = \tan(x)$.
- Use the derivative rule for tangent
The derivative of the tangent function is found using the formula: $$ \frac{d}{dx}(\tan(x)) = \sec^2(x) $$
- Apply the derivative rule
Since our function is $f(x) = \tan(x)$, we apply the derivative rule: $$ f'(x) = \sec^2(x) $$
The derivative of the tangent function is $f'(x) = \sec^2(x)$.
More Information
The derivative of the tangent function is a fundamental result in calculus. It shows how the slope of the tangent line to the curve changes at any point on the function. An interesting fact is that the secant function, $\sec(x)$, is itself related to the cosine function, as it is defined as the reciprocal of cosine, $\sec(x) = \frac{1}{\cos(x)}$.
Tips
- Forgetting the secant squared formula: Make sure to remember the derivative formula, as it is specific to the tangent function.
- Confusion with derivatives of similar functions: Be careful to differentiate not only tangent but also sine and cosine functions correctly.
