show me how to find the equation to the tangent line to y=tan(x) at x=0

To find the equation of the tangent line to the curve \( y = \tan(x) \) at \( x = 0 \), follow these steps:

### 1. **Find the value of \( y \) at \( x = 0 \)**:
We need to evaluate \( y = \tan(x) \) at \( x = 0 \).
\[
y = \tan(0) = 0
\]
So, the point of tangency is \( (0, 0) \).

### 2. **Find the derivative of \( y = \tan(x) \)**:
The derivative of \( \tan(x) \) with respect to \( x \) gives the slope of the tangent line at any point:
\[
\frac{d}{dx}[\tan(x)] = \sec^2(x)
\]
Now, evaluate this derivative at \( x = 0 \):
\[
\sec^2(0) = 1
\]
Thus, the slope of the tangent line at \( x = 0 \) is \( 1 \).

### 3. **Write the equation of the tangent line**:
Using the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope, and \( (x_1, y_1) \) is the point of tangency. In this case, \( m = 1 \), \( x_1 = 0 \), and \( y_1 = 0 \). Substituting these values into the point-slope equation:
\[
y - 0 = 1(x - 0)
\]
Simplifying:
\[
y = x
\]
So, the equation of the tangent line to \( y = \tan(x) \) at \( x = 0 \) is:
\[
y = x
\]

Would you like more details on the steps or need clarification?

### Related Questions:
1. How do you find the tangent line of other trigonometric functions, like \( \sin(x) \)?
2. How would the tangent line change if the function was \( y = \tan(2x) \)?
3. What is the derivative of the secant function, \( \sec(x) \)?
4. How do you find the tangent line to any curve at a specific point?
5. What happens to the slope of the tangent line for \( y = \tan(x) \) at different values of \( x \)?

**Tip:** When finding tangent lines, always remember that the derivative gives you the slope at a point.

Learn how to find the equation of the tangent line to the function y = tan(x) at x = 0. This step-by-step solution covers key calculus concepts, including finding the derivative and using the point-slope formula to calculate the tangent line. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation