In Geometry, a straight line which touches a plane curve at a given point is called the tangent line to the curve or just tangent. Tangent line always just touches the surface of the curve at a point. The point at which it touches the curve is called the point of tangency.
Tangent Line


In Calculus, the tangent line is a straight line to a given curve  ${y = f(x)}$  at a point  $x_{0}$ , which passes through the point  $(x_{0}, f(x_{0}))$ on the curve and its slope is given by  $f'(x_{0})$, where $f'(x)$ represents the derivative of the function $f(x)$.

Steps to find the slope of a tangent and equation of tangent line :
Step 1: Identify the given equation and the given point at which the tangent equation has to be defined
Step 2: Write the equation in the form of   ${y = f(x)}$.
Step 3: Differentiate the equation  ${y = f(x)}$ to get $\frac{dy}{dx}$ $= f'(x)$.

Step 4:
Plug the given point $x_{0}$ in the derivative i.e., $\frac{dy}{dx}$ $= f'(x_{0})$.
Step 5: The slope of the tangent, $m = f'(x_{0})$
Step 6: Use the point slope form  $y - y_{0} = m(x - x_{0})$ and plug the values of $x_{0}, y_{0}$
Step 7: The obtained equation is the equation of line.

The slope of a tangent line parallel to $x$ axis is zero and the slope of the tangent line which is parallel to $y$ – axis is not defined.