Differentiation rules > Chain rule
123456Chain rule

Exercises

Exercise 1

Differentiate the following functions.

a

f ( x ) = ( x 2 - 100 ) 4

b

g ( x ) = -5 + ( 1 - x ) 3

c

H ( t ) = 25 ( 2 - 4 t ) 3

d

y ( x ) = 2 p 2 x - ( p x + 3 ) 4

Exercise 2

Here you see the graph of the function f ( x ) = - ( 2 x - 6 ) 3 + 4 .

a

The graph seems to be decreasing for every value of x except x = 3 . Prove that this is indeed the case.

b

The tangent to the graph of f for x = 2 intersects the x -axis in the point P . Calculate the coordinates of P .

Exercise 3

Find the derivatives of the following functions:

a

y = x 7 3

b

f ( x ) = 1 x 3 + 4 x 2 - 3 x + 1

c

H ( p ) = ( 1 - p ) 3

d

g ( x ) = 2 x - 5 1 - x

Exercise 4

Here you see the graph of the function f ( x ) = x + 8 - x 2 .

a

Determine the domain of f .

b

Algebraically compute the range of f .

c

Denote the edgepoints of the graph of f by A and B . For what value of x is the slope of the tangent to the graph f equal to that of the line A B ?

Exercise 5

A water line needs to be laid from point A to point C. Along the street the cost is € 30,00 per meter and though the field € 70,00 per meter. The length of A B is 600 meter and the length of B C is 500 meter. There are several ways to lay the water line:

  • along the street to point B and then through the bordering terrain to point C ;

  • directly from A through the field, in a straight line to C ;

  • or in one of many ways in between: the line then is laid partly along the street to point D , and then from the street to point C .

a

What is the cost if you choose the first option?

b

What is the cost if you choose the second option?

c

Study the third option. Denote the length of B D by the variable x . Now express the cost voor the laying of this water line in x .

d

How should the water line be laid to minimize the cost? Calculate the minimal cost using the derivative.

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