Determine Derivatives of Composite Functions?

Question

Steps involved in determining these?

A. f(x)=sin(4x)

B. f(x)=(x+3)(x^3-x+5)

C. f(x)=In(x)/(x^3-5x)

Answer 1

A) The function is .

Derivating on both sides we get,

Derivative of sinx is cosx.

Derivative of ax = a.

is the answer.

 

Answer 2

• B).

The function is f (x) = (x + 3)(x³ - x + 5).

Apply derivative on each side.

f ' (x) = d/dx [ (x + 3)(x² - x + 5) ].

Apply product rule : ( uv ) ' = uv ' + vu '.

f ' (x) = { (x + 3) * [d/dx (x³ - x + 5)] } + { (x³ - x + 5) * [d/dx (x + 3)] }.

= [ (x + 3) * (3x² - 1) ] + [ (x³ - x + 5) * 1 ]

= [ 3x³ + 9x² - x - 3] + [ x³ - x + 5 ]

= 3x³ + 9x² - x - 3 +  x³ - x + 5

= 4x³ + 9x² - 2x + 2.

∴ f ' (x) = 4x³ + 9x² - 2x + 2.

Answer 3

• C).

The function is .

Apply derivative on each side.

Apply quotient rule : .

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