3 Extending the table of derivatives

We now quote simple rules which enable us to extend the range of functions which we can differentiate. The first two rules are for differentiating sums or differences of functions. The reader should note that all of the rules quoted below can be obtained from first principles using the approach outlined in Section 11.1.

Key Point 5
Rule 1:       The derivative of f ( x ) + g ( x ) is d f d x + d g d x
Rule 2:       The derivative of f ( x ) − g ( x ) is d f d x − d g d x

These rules say that to find the derivative of the sum (or difference) of two functions, we simply calculate the sum (or difference) of the derivatives of each function.

Example 3

Find the derivative of y = x 6 + x 4 .

Solution

We simply calculate the sum of the derivatives of each separate function:

d y d x = 6 x 5 + 4 x 3

The third rule tells us how to differentiate a multiple of a function. We have already met and applied particular cases of this rule which appear in Table 1.

Key Point 6
Rule 3:       If  k  is a constant, the derivative of k f ( x ) is k d f d x

This rule tells us that if a function is multiplied by a constant, k , then the derivative is also multiplied by the same constant, k .

Example 4

Find the derivative of y = 8 e 2 x

Solution

Here we are interested in differentiating a multiple of the function e 2 x . We differentiate e 2 x , giving 2 e 2 x , and multiply the result by 8. Thus

d y d x = 8 × 2 e 2 x = 16 e 2 x

Example 5

Find the derivative of   y = 6 sin 2 x + 3 x 2 − 5 e 3 x

Solution

We differentiate each part of the function in turn.

y = 6 sin 2 x + 3 x 2 − 5 e 3 x d y d x = 6 ( 2 cos 2 x ) + 3 ( 2 x ) − 5 ( 3 e 3 x ) = 12 cos 2 x + 6 x − 15 e 3 x
Task!

Find d y d x where y = 7 x 5 − 3 e 5 x .

First find the derivative of 7 x 5 :

7 ( 5 x 4 ) = 35 x 4

Next find the derivative of 3 e 5 x :

3 ( 5 e 5 x ) = 15 e 5 x

Combine your results to find the derivative of 7 x 5 − 3 e 5 x :

35 x 4 − 15 e 5 x

Task!

Find d y d x where y = 4 cos x 2 + 17 − 9 x 3 .

First find the derivative of 4 cos x 2 :

4 ( − 1 2 sin x 2 ) = − 2 sin x 2

Next find the derivative of 17:

0 Then find the derivative of − 9 x 3 :

3 ( − 9 x 2 ) = − 27 x 2

Finally state the derivative of y = 4 cos x 2 + 17 − 9 x 3 :

− 2 sin x 2 − 27 x 2

Exercises
  1. Find d y d x when y is given by:

    (a)   3 x 7 + 8 x 3   (b)   − 3 x 4 + 2 x 1.5   (c)   9 x 2 + 14 x − 3 x   (d)   3 + 2 x 4   (e)   ( 2 + 3 x ) 2

  2. Find the derivative of each of the following functions:
    1. z ( t ) = 5 sin t + sin 5 t   
    2.   h ( v ) = 3 cos 2 v − 6 sin v 2
    3.   m ( n ) = 4 e 2 n + 2 e 2 n + n 2 2   
    4.   H ( t ) = e 3 t 2 + 2 tan 2 t   
    5.   S ( r ) = ( r 2 + 1 ) 2 − 4 e − 2 r
  3. Differentiate the following functions.
    1.   A ( t ) = ( 3 + e t ) 2
    2.   B ( s ) = π e 2 s + 1 s + 2 sin π s
    3.   V ( r ) = ( 1 + 1 r ) 2 + ( r + 1 ) 2
    4.   M ( θ ) = 6 sin 2 θ − 2 cos θ 4 + 2 θ 2
    5.   H ( t ) = 4 tan 3 t + 3 sin 2 t − 2 cos 4 t
    1. 21 x 6 + 24 x 2
    2.   − 12 x 3 + 3 x 0.5
    3.   − 18 x 3 − 14 x 2 − 3
    4. 1 2   
    5. 12 + 18 x
    1. z ′ = 5 cos t + 5 cos 5 t   
    2. h ′ = − 6 sin 2 v − 3 cos v 2   
    3.   m ′ = 8 e 2 n − 4 e − 2 n + n
    4.   H ′ = 3 e 3 t 2 + 4 sec 2 2 t   
    5. S ′ = 4 r 3 + 4 r + 8 e − 2 r
    1.   A ′ = 6 e t + 2 e 2 t   
    2. B ′ = 2 π e 2 s − 1 s 2 + 2 π cos ( π s )
    3.   V ′ = − 2 r 2 − 2 r 3 + 2 r + 2   
    4.   M ′ = 12 cos 2 θ + 1 2 sin θ 4 + 4 θ
    5. H ′ = 12 sec 2 3 t + 6 cos 2 t + 8 sin 4 t