The derivative of a function f(x) with respect to x, written as , is defined as:

, if this limit exists.

We read as: **f prime of x** or as **f dash x**.

**Notation: **There are many ways to denote the derivative of a function f(x) with respect to x.

Besides , the most common notations are:

D_{x}f(x) and f(x)

If the function is of the form y = f(x), then the most common notations for the derivative are:

y', and D_{x}y

**Derivative of a function at a number: **

The derivative of a function f(x) with respect to x at a number x = c, written as , is defined as:

, if this limit exists.

A function is said to be derivable or differentiable at x = c, if it has a derivative there.

A function is said to be differentiable on an interval, if it is differentiable at every point of the interval.

## Interpretation of the Derivative as the Slope of a Tangent

If a curve C has equation y = f(x) and we want to find the tangent to C at the point P(c, f(c)), then we consider a nearby point Q(c + h, f(c + h)), where h ≠ 0, and compute the slope of the secant line PQ:

m_{PQ} =

Then we let Q approach P along the curve C by letting h approach 0. If m_{PQ} approaches a number m, then we define tangent **t** to be the line through P with slope m. This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P.

The tangent line to the curve y = f(x) at the point P(c, f(c)) is the line through P with slope

m = , provided that this limit exists.

Since, by the definition of a derivative of the function, this is the same as the **derivative** .

Thus, the tangent line to the curve y = f(x) at the point P(c, f(c)) is the line through (c, f(c)) whose slope is equal to , the derivative of f(x) at c.

If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve y = f(x) at the point (c, f(c)): **y – f(c) = (x – c).**

Suppose 'y' is a quantity that depends on another quantity 'x'. Thus, 'y' is a function of 'x' and we write y = f(x).

If 'x' changes from x_{1} to x_{2}, then the **change** in 'x', also called the **increment** of 'x', is: Δx = x_{2} – x_{1}

and the corresponding change in 'y' is: Δy = f(x_{2}) – f(x_{1}).

The difference quotient:

is called the **average rate of change of 'y' w. r. t. 'x'** over the interval [x_{1}, x_{2}].

It can be interpreted as the slope of the secant line PQ in below figure.

Consider the average rate of change over smaller and smaller intervals by letting x_{2} approach x_{1} and therefore letting Δx approach 0. The limit of these average rates of change is called the instantaneous rate of change of 'y' with respect to 'x' at x = x_{1}, which is interpreted as the slope of the tangent to the curve y = f(x) at P(x_{1}, f(x_{1})).

Thus, instantaneous rate of change = ⋅⋅⋅⋅⋅⋅ (i)

From equation (i), we recognize this limit as being the derivative of f(x) at x_{1}, that is, f '(x_{1}). Therefore, the derivative is the instantaneous rate of change of y = f(x) with respect to 'x' when x = c.

**Ex:** The position of a particle is given by the equation of motion s = f(t) = t^{2} – 2t + 2, where 's' is measured in meters and 't' in seconds. Find the velocity after 3 seconds.**Sol:**

L'Hospital's Rule for indeterminate forms |
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• If f(x) and g(x) be two functions of 'x' such that f(x) = 0 and g(x) = 0, then |

• The above result is also applicable if f(x) = ± ∞ and g(x) = ± ∞ |

• If assumes the indeterminate form and satisfies all the conditions embodied in L'Hospital's Rule, we can repeat the application of this rule on to get Some times it may be necessary to repeat this process a number of times till our goal of evaluating limit is achieved. |