1. Equation of the Tangent Line
1. Slope of the Tangent Line
   When the function is differentiable at , the slope of the tangent line at the point on the curve is equal to the derivative at .
2. Equation of the Tangent Line
   When the function is differentiable at , the equation of the tangent line at the point on the curve is:
   
   
   
   
   
   The equation of a line passing through the point on the curve and perpendicular to the tangent line at this point is:
   
3. Steps to Find the Equation of a Tangent Line
1. Tangent Line at a Point on the Curve
1. Find the slope of the tangent line .
2. Substitute into the equation .
2. Tangent Line with a Given Slope to the Curve
1. Set the coordinates of the point of tangency as .
2. Use the condition to find .
3. Substitute the value of into the equation .
3. Tangent Line from a Point Outside the Curve
1. Set the coordinates of the point of tangency as .
2. Find the slope of the tangent line .
3. Use the fact that the line passes through the point to find .
4. Substitute the value of into the equation .


4. Common Tangent Line
   If two curves and have a common tangent line at , then:
   
   
   
   
   This means:
1. The two curves meet at the point .
   
2. The slopes of the tangent lines to both curves at are the same.