What is a Derivative?

The derivative is defined as an instantaneous rate of change of a function at any given point. This instantaneous rate of change is also known as a slope. The general notation of a derivative function is given by the following function:

= \frac {\triangle d(y)} {\triangle d(x)}

In other words, we can say that a derivative of a function is the rate of change in y with respect to x. The integration of a function is inverse of the derivative. The process of calculating a derivative is known as differentiation. In this article, we will learn some of the derivative rules or rules of differentiation along with the relevant examples.

Derivative of a Constant

The derivative of a constant function is equal to zero.

Example

If a is a constant, then the following function will have the zero derivative:

f(x) = a

f ' (x) = 0

Here, we have represented derivative of  function by f ' (x). We can also represent a derivative function by \frac {d}{d(x)} (c). Both notations are used to represent the derivative of a function.

Derivative of a Constant Multiple

If we have to find the derivative of a function in which the constant is multiplied by the function, then we take THE derivative of the function and multiply the constant with it. The general notation of this rule is given below:

If f(x) = mx, then f '(x) = m x'

Example 1

Find the derivative of the function f(x) = 7 x ^ 3.

Solution

In the above function, a constant 7 is multiplied by the variable x ^3. So, first we will write 7 as it is and take the derivative of the cubic variable x ^3 inside the parentheses.

f ' (x) = 7 ( 3 x ^ 2)

To take the derivative of the cubic function x ^ 3, we have used the power rule which says that if f(x) = x ^n, then f ' (x) = n x ^ {n - 1}.

Now, multiply 7 with the term inside the brackets to get the final value of f ' (x).

f ' (x) = 21 x ^ 2

Example 2

Find the derivative of the function f(x) = \frac {x ^2}{6}.

Solution

You can notice that in the above function, the fraction {1}{6} is multiplied by the squared variable x ^2. Hence, we can write the function as a product of fraction and variable like this:

f  (x) = \frac {1}{6} (x ^ 2)

Now, use the derivative power rule x ^ n = n x ^ {n - 1} to get the derivative of the function inside parentheses:

f ' (x) = \frac {1}{6} (2x)

We will get the following final answer after simplifying the above derivative:

f ' (x) = \frac {x} {3}

Derivative of x

The derivative of x is 1 because when we will apply the power rule, we will get the exponent 0 of the variable x. Since, x ^ 0 = 1, hence as a general rule we can say that the derivative of x is 1. Mathematically, it can be written as:

If f(x) = x, then f ' (x) = 1

Derivative Power Rule

We have already used this rule while differentiating the functions in the above examples. The power rule of derivative is expressed as:

If f(x) = x ^ n, then f ' (x) = n x ^ {n - 1}

Example

Find the derivative of the function f (x) =  x ^ {10}.

Solution

We will simply apply the derivative power rule to find the derivative of the above function. Since, in the above function the value of n is 10, hence the derivative of the function will be written as follows using this value of n:

f ' (x) = 10 x ^ {10 - 1}

f ' (x) =10 x ^ 9

Derivative of a Root

The derivative rule of the root is explained below mathematically:

If f (x) = \sqrt {m}, then f ' (x) = \frac { m ' }{2 \cdot \sqrt {m}}

The above formula is for the derivative of a function which involves a square root. If you have to find the derivative of a root function other than square root, then the following formula should be used:

If f(x) = \sqrt [k] {u}, then f ' (x) = \frac { u ' } {k \cdot \sqrt [k] { u ^ {k - 1}}}

Let us explain the above rule by the following examples.

Example 1

Find the derivative of the function f(x) = \sqrt {2x}

Solution

The derivative of the term inside the root symbol is 6. Using the derivative square root formula f ' (x) = \frac { m ' }{2 \cdot \sqrt {m}}, we will write the final answer as follows:

f '  (x) = \frac {2} {2 \cdot \sqrt {2x}}

To simplify the above answer, we will simply divide the numerator and the denominator by 2:

f ' (x) = \frac {1} {\sqrt {2x}}

Example 2

Find the derivative of the function f(x) = \sqrt [3] {x}

Solution

To find the derivative of the above function, use the formula f ' (x) = \frac { u ' } {k \cdot \sqrt [k] { u ^ {k - 1}}}.

f ' (x) = \frac {1} { 3 \cdot \sqrt [3] {x ^ {1 - 1}}}

f ' (x) = \frac {1} { 3 \cdot \sqrt [3] {x ^2}}

It can also be written as:

f ' (x) = \frac {1}{3 x ^ {\frac {2}{3}}}

 

Derivative of a Sum or Difference

The derivative of a sum or difference of two numbers is equal to the derivatives of individual terms. If m and n are two terms, then their derivative will be calculated as follows:

If f (x) = m \pm n, then f ' (x) = m ' \pm n '

Example 1

Find the derivative of the function f(x) = x ^ 2 + 4x.

Solution

To find the derivative of the above function, we will use the derivative sum rule which works by finding the derivative of the individual terms involved in it. The derivative of x ^ 2 is 2x which is calculated using the power rule and the derivative of 4x is 4. Hence, the derivative of the entire function is given below:

f ' (x) =  2x + 4

Example 2

Find the derivative of the function f(x) = x ^ 3 - 5x.

Solution

Find the derivatives of the individual terms involved in the function. The derivative of x ^ 3 is 3 x^ 2 and the derivative of 5x is 5.

f ' (x) = 3 x ^ 2 - 5

Derivative Product Rule

The mathematical notation of the derivative product rule is given below:

If f (x) = m \cdot n, then f ' (x) = m' \cdot n + m \cdot n '

The above rule is explained by the following example.

Example

Find the derivative of the function f(x) = x ^ 4 \cdot 5x.

Solution

Find the derivative of individual factors involved in the above function. The derivative of x ^ 4 is 4 x ^ 3 and the derivative of 5x is 5. Hence, using the formula f (x) = m \cdot n, then f ' (x) = m' \cdot n + m \cdot n ', we will write the derivative of the whole function like this:

f ' (x) = 4 x ^ 3 \cdot 5x + x ^ 4 \cdot 5

f ' (x) = 20 x ^ 4 + 5 x ^ 4

f ' (x) = 25 x ^ 4

Derivative Quotient Rule

The derivative quotient rule is expressed mathematically as:

If f(x) = \frac {u}{v}, then f ' (x) = \frac { u ' \cdot v - u \cdot v'}{v ^ 2}

Example

Find the derivative of the function f(x) = \frac {9 x ^ 2}{2x}.

Solution

The derivative of the term 9 x ^ 2 is 18x and the derivative of the term 2x is equal to 2. We will put these values in the quotient derivative formula which says that if f(x) = \frac {u}{v}, then f ' (x) = \frac { u ' \cdot v - u \cdot v'}{v ^ 2}:

f ' (x) = \frac { 18x \cdot 2x - 9x ^ 2 \cdot 2} { 4 x ^ 2}

f ' (x) = \frac { 36 x ^ 2 - 18 x ^ 2} { 4x ^ 2}

f ' (x) = \frac { 18 x ^ 2} { 4x ^ 2}

f ' (x) = \frac {9x ^ 2} { 2x ^ 2}

f ' (x) = \frac {9}{2}

Derivative Reciprocal Rule

The derivative reciprocal rule is mathematically denoted as follows:

If f(x) = \frac {1} { v}, then f ' (x) = \frac { -v'} { v ^ 2}

Example

Find the derivative of the function f (x) = \frac {1} { 4x ^ 2}

Solution

Since, the above function is in the reciprocal form, therefore we will use the derivative formula f ' (x) = \frac { -v'} { v ^ 2} to find the derivative. The derivative of the function 4 x ^ 2 is 8x. Put this value in the formula to get the derivative:

f ' (x) = \frac { - 8x } { 16 x ^ 4}

Divide the the terms in the numerator and the denominator by - 8x:

f ' (x) = - \frac {1} { 2x ^ 3}

Derivative of an Exponential Function

The derivative of the exponential functions is mathematically denoted as:

If f (x) = a ^ u, then f ' (x) = u ' \cdot a ^ u \cdot ln a

Example

Find the derivative of the exponential function f (x) = 5 ^ {6x + 1}.

Solution

We will find the derivative of the above function in four steps:

Step 1 - First write the function as it is f (x) = 5 ^ {6x + 1}

Step 2 - Find the derivative of the function in power. The derivative of the function in the exponent is a linear equation. We know that the the derivative of a linear function is equal to its slope,  hence its derivative will be equal  6.

Step 3 - Find the logarithm of the base. The base in the above function is 5, so its log will be written as ln 5.

Step 4 - We will get the derivative of the whole function by multiplying the terms obtained in step 1, 2 and 3:

f ' (x) = 5 ^ {6x + 1} \cdot 6 \cdot ln 5

Derivative of a Logarithmic Function

The mathematical notation of the derivative of the logarithmic functions is given below:

If f(x) = log _a u, then f ' (x) = \frac {u '}{u \cdot lna}

Example

Find the derivative of the logarithmic function f (x) = log _ 5 7x.

Solution

First, find the derivative of 7x. The derivative of 7x is 7. Put this value in the formula f ' (x) = \frac {u '}{u \cdot lna}.

f ' (x) = \frac {7} {7x ln 5}

f ' (x) = \frac {1}{ x ln 5}

Derivative of a Sine Function

The mathematical notation of this derivative rule is given below:

If f (x) = sin u, then f ' (x) = u ' \cdot cos u

Example

Find the derivative of the function f (x) = sin 6x

Solution

The derivative of 6x is 6. Put this value in the formula f ' (x) = u ' \cdot cos u.

f ' (x) = 6 cos 6x

Derivative of a Cosine Function

The derivative of a cosine function is mathematically denoted as follows:

If f (x) = cos u , then f ' (x) = - u' \cdot sin u

Example

Find the derivative of the function f (x)  = cos 7x

Solution

The derivative of the term 7 x is 7. Putting this value in the formula we will get:

f ' (x) = - 7 \cdot sin (7x)

Derivative of a Tangent Function

The derivative of a tangent function is mathematically denoted as follows:

If f(x) = tan x, then f ' (x) = \frac { u'}{cos ^2 u} or u' \cdot sec ^2 u

In the above formula, cos represents cosine function and sec represent secant function.

Example

Find the derivative of the tangent function f(x) = tan 3x.

Solution

The derivative of the term 3x is 3. Put this value in the tangent formula to get the following answer:

f ' (x) = \frac { 3} {cos ^2 3x} or 3 \cdot sec ^2 3x

Derivative Chain Rule

We find the derivative chain rule to find the derivative of a composite function. We use the derivative chain rule when we have a function inside another function. Mathematically the derivative chain rule is denoted as follows:

If there is a f (g (x)), then its derivative is f ' (g (x) ) g ' (x)

Example

Find the derivative of the function f (x) = (2x ^2 + 7) ^ 3 .

Solution

We cannot solve the above function using any of the above differentiation rules except the chain rule because this is a composite function. There is a function inside another function. First, we have a function 2x ^2 + 7 and then we are taking the cube of the entire function. We can say that f(x) = x ^ 3 and g (x) = 2x ^2 + 7. When we will write the function as f ( g (x) ), we will get the function (2x ^2 + 7) ^ 3.

Using the derivative chain rule, we will write the derivative of the function as follows:

f ' (x) = 3 (2 x ^2 + 7) ^ 2 \cdot 4x

We can rewrite it after simplification as follows:

f ' (x) = 12x (2x ^2 + 7) ^ 2

Did you like the article?

1 Star2 Stars3 Stars4 Stars5 Stars (1 votes, average: 5.00 out of 5)
Loading...

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

Did you like
this resource?

Bravo!

Download it in pdf format by simply entering your e-mail!

{{ downloadEmailSaved }}

Your email is not valid