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Equation of the Tangent Line

If a function is differentiable at , then the slope of the tangent line at the point on the curve is . Therefore, the equation of the tangent line is:
The equation of the line perpendicular to the tangent at the point is: provided

Methods to Find the Equation of the Tangent Line

  1. At a point on the curve
    1. Find the slope of the tangent line, .
    2. Substitute into to find the equation of the tangent line.

  2. When the tangent line has a given slope
    1. Set the coordinates of the point of tangency as .
    2. Use to find the coordinates of the point of tangency.
    3. Substitute into to find the equation of the tangent line.

  3. From an external point
    1. Set the coordinates of the point of tangency as .
    2. Find the slope of the tangent line, .
    3. Substitute into , then substitute the coordinates to find the value of .
    4. Substitute into to find the equation of the tangent line.

  4. For a parametric curve at
    1. Find .
    2. Find the values of and .
    3. Substitute into to find the equation of the tangent line.

  5. For the curve at the point
    1. Use the implicit differentiation method to find .
    2. Substitute the coordinates of into to find the slope of the tangent line.
    3. Use the coordinates of and the slope from (ii) to find the equation of the tangent line.

  6. If two curves and share a common tangent line at , then:
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