**Differentiation** is a method to compute the rate at which a dependent output y changes
with respect to the change in the independent input x. This rate of change is called
the

**derivative** of y with respect to x.

In other words,

**The process of finding a derivative is called Differentiation.**

The graph of a function, drawn in blue, and a tangent line to that function, drawn in green.
The slope of the tangent line is equal to the derivative of the function at the marked point.

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**Derivative of a constant function**

The derivative of f(x) = c where c is a constant is given by

## f '(x) = 0

**Example: **f(x) = - 10 , then f '(x) = 0

**Derivative of a power function (Power rule)**

The derivative of f(x) = x

^{r} where r is a constant real number is given by

## f '(x) = r *x^{(r-1)}

**Example: ** f(x) = x

^{-2} , then f '(x) = -2 x

^{-3} = -2 / x

^{3}
**Derivative of a function multiplied by a constant**

The derivative of f(x) = c g(x) is given by

## f '(x) = c g '(x)

**Example:** f(x) = 3x

^{3} ,

let c = 3 and g(x) = x

^{3}, then f '(x) = c g '(x)
= 3 (3x

^{2}) = 9 x

^{2}
**Derivative of the sum of functions (Sum rule)**

The derivative of f(x) = g(x) + h(x) is given by

## f '(x) = g '(x) + h '(x)

**Example:**
f(x) = x

^{2} + 4

let g(x) = x

^{2} and h(x) = 4, then f '(x) = g '(x) + h '(x) = 2x + 0 = 2x

**Derivative of the difference of functions**

The derivative of f(x) = g(x) - h(x) is given by

## f '(x) = g '(x) - h '(x)

**Example:**
f(x) = x

^{3} - x

^{-2}
let g(x) = x

^{3} and h(x) = x

^{-2}, then

f '(x) = g '(x) - h '(x) = 3 x

^{2} - (-2 x

^{-3}) = 3 x

^{2} + 2x

^{-3}
** Derivative of the product of two functions(Product rule)**

The derivative of f(x) = g(x) h(x) is given by

## f '(x) = g(x) h '(x) + h(x) g '(x)

**
**

Example:
f(x) = (x

^{2} - 2x) (x - 2)

let g(x) = (x

^{2} - 2x) and h(x) = (x - 2), then

f '(x) = g(x) h '(x) + h(x) g '(x) = (x

^{2} - 2x) (1) + (x - 2) (2x - 2)
= x

^{2} - 2x + 2 x

^{2} - 6x + 4

=3 x

^{2} - 8x + 4

**Derivative of the quotient of two functions (Quotient rule)**

The derivative of f(x) = g(x) / h(x) is given by

## f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x)^{2}

**
Example:** f(x) = (x - 2) / (x + 1)

let g(x) = (x - 2) and h(x) = (x + 1), then

f '(x) = ( h(x) g '(x) - g(x) h '(x) ) / h(x)

^{2}
= ( (x + 1)(1) - (x - 2)(1) ) / (x + 1)

^{2}
= 3 / (x + 1)

^{2}
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An equation of the form f(x, y) in which y is not expressible directly in terms of x, is known as an implicit function of x and y.

To implicitly differentiate an implicit function follow the steps below.

### 1. Apply all differentiation rules. Except to y.

Given equation is:

y

^{3} - x

^{2} = -5

y3 - 2x = 0

### 2. Then differentiate the y's in regard to x. In turn you add a y prime
to the differentiated y's

3y

^{2}y' -2x = 0

Here, y' is refered to d(y)/d(x)

### 3. Now factor out any y primes

Hint: In this example there are none.

### 4. Lastly, place all the y primes on one side

Solution:

**y' = (2x) / (3y**^{2})
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