Math 22 Product Rule and Quotient Rule

The Product Rule

 The derivative of the product of two differentiable functions is equal to the first function times 
 the derivative of the second plus the second function times the derivative of the first.
 ${\frac {d}{dx}}[f(x)\cdot g(x)]=f'(x)g(x)+f(x)g'(x)$

Example: Find derivative of

1) $f(x)=(x+1)(x^{2}+3)$

Solution:
$f'(x)={\frac {d}{dx}}[(x+1)](x^{2}+3)+(x+1){\frac {d}{dx}}[(x^{2}+3)]$
$=(1)(x^{2}+3)+(x+1)(2x)=x^{2}+3+2x^{2}+2x=3x^{2}+2x+3$

2) $f(x)=(4x+3x^{2})(6-3x)$

Solution:
$f'(x)={\frac {d}{dx}}[(4x+3x^{2})](6-3x)+(4x+3x^{2}){\frac {d}{dx}}[(6-3x)]$
$=(4+6x)(6-3x)+(4x+3x^{2})(-3)=24+36x-12x-18x^{2}-12x-9x^{2}=-27x^{2}+12x+24$

3) $f(x)=(e^{2}+x^{2})(4x+5)$

Solution:
$f'(x)={\frac {d}{dx}}[(e^{2}+x^{2})](4x+5)+(e^{2}+x^{2}){\frac {d}{dx}}[(4x+5)]$
$=(2x)(4x+5)+(e^{2}+x^{2})(4)=8x^{2}+10x+4e^{2}+8x^{2}=-16x^{2}+10x+4e^{2}$

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