Math 230/ W3DQS

Q1 . Writing the given functions as composition of two functions

f (x (3x 2 )2 g (h (x

where h (x 3x 2 and g (x x2

Similarly , f (x 2x2 - 7 g (h (x

Where , h (x x2 - 7 /2 and g (x 2x

Other examples of composite function are

f (x 1 (1 x )2 g (h (x

where h (x (1 x )2 and g (x 1 /x

Also , f (x 1 /Cosx g (h (x

Where h (x Cosx and g (x 1 /x

Q (2

) Derivative of a linear combination of two functions is the same linear combination of the derivatives of the individual functions

Example 1

Let us consider two functions g (x x2 and h (x 5x 2 Now , let us consider a function f (x ) which is a linear combination of g (x ) and h (x

i .e . f (x a g (x b h (x ax2 5bx 2b where , `a ' and `b ' are constants Thus it can be seen that the derivative of a linear combination of two functions is the same linear combination of the derivatives of the individual functions

Now let us take another example

Example 2

Let us consider two functions g (x Sinx and h (x Cosx Now , let us consider a function f (x ) which is a linear combination of g (x ) and h (x

i .e . f (x a g (x b h (x aSinx bCox where , `a ' and `b ' are constants Thus it can be seen that the derivative of a linear combination of two functions is the same linear combination of the derivatives of the individual functions

Q3 . Applying quotient rule for determining the derivative of the ratio of two functions Now quotient rule says if a function f (x ) can be represented a ratio of two functions g (x ) and h (x ) such that f (x g (x /h (x then its derivative is given as...