f'(0)

Calculus Level pending

I f f ( x ) = 2 x + 3 x 2 + 1 , t h e n f ′ ( 0 ) = ? If\quad f(x)=\frac { 2x+3 }{ { x }^{ 2 }+1 } ,\\ then\quad f'(0)=?


The answer is 2.

Justin Tuazon by Justin Tuazon , 33, Philippines

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1 solution

Luis Tabule
Dec 7, 2014

This is my solution for the math problem. f ( x ) = 2 x + 3 x 2 + 1 L e t : u = 2 x + 3 v = x 2 + 1 B y A p p l y i n g t h e D e r i v a t i v e s o f R a t i o n a l F u n c t i o n s R u l e : f ′ ( x ) = u ′ v − u v ′ v 2 f ′ ( x ) = 2 ( x 2 + 1 ) − ( 2 x + 3 ) ( 2 x ) ( x 2 + 1 ) 2 f ′ ( x ) = 2 − 2 x 2 + 6 x x 4 + 2 x 2 + 1 f ′ ( 0 ) = 2 1 = 2 f\left( x \right) =\frac { 2x+3 }{ { x }^{ 2 }+1 } \\ Let:\\ u=\quad 2x+3\\ v={ \quad x }^{ 2 }+1\\ \\ By\quad Applying\quad the\quad Derivatives\quad of\quad Rational\quad Functions\quad Rule:\\ f'\left( x \right) =\quad \frac { u'v-uv' }{ { v }^{ 2 } } \\ f'(x)=\quad \frac { 2({ x }^{ 2 }+1)-(2x+3)({ 2x) } }{ { ({ x }^{ 2 }+1) }^{ 2 } } \\ f'(x)=\quad \frac { 2-{ 2x }^{ 2 }+6x }{ { x }^{ 4 }+{ 2x }^{ 2 }+1 } \\ f'(0)=\quad \frac { 2 }{ 1 } =\quad 2

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