__Geometrical interpretation of derivative :__

Consider a function y = f(x) and points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) on it. As x_{1} changes to x_{2}, y_{1} becomes y_{2}. Average rate of change will be given by

(which is clearly the slope of line PQ).

As Q -> P i.e. as x_{2} -> x_{1} or Δx -> 0,

the average rate of change Δy/Δx becomes the instantaneous rate of change represented by dy/dx and thus dy/dx represent the slope of the tangent at P.

__Tangent and Normal__

The derivative of a function y = f(x) represents the slope of the tangent to the curve at the general point (x, y).

__Equation of Tangent and Normal:__

Let y = f (x) be the given curve. We already know that dy/dx at any point lying on the curve would give us the slope of the tangent that can be drawn at that point.

Let (x_{1}, y_{1}) be any point on the curve, that means, y_{1} = f(x_{1}).

Now the slope of the tangent that can be drawn to the curve at (x_{1}, y_{1}) will be

Thus the equation of the tangent at (x_{1}, y_{1}) would be,

Similarly, the equation of the normal at (x_{1}, y_{1}) would be

provided that

**Note:** If x = g (t) , y = h (t)

then ,

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