C h a p t e r 3 S e c t i o n 2 {\displaystyle \mathbf {Chapter3Section2} }
Review section 1
d d x [ c ] = 0 {\displaystyle {\frac {d}{dx}}[c]=0}
d d x [ c ⋅ f ( x ) ] = c ⋅ d d x [ f ( x ) ] {\displaystyle {\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]}
d d x [ f ( x ) ± g ( x ) ] = d d x [ f ( x ) ] ± d d x [ g ( x ) ] {\displaystyle {\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]}
d d x [ a x ] = ln ( a ) a x {\displaystyle {\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}}
d d x [ e x ] = e x {\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}}
P o w e r R u l e {\displaystyle \color {green}Power\,Rule}
d d x [ x n ] = n ⋅ x n − 1 {\displaystyle {\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1}
N e w r u l e s f o r 3.2 {\displaystyle \mathbf {New\ rules\ for\ 3.2} }
P r o d u c t R u l e {\displaystyle \color {Blue}Product\,Rule}
d d x [ f ⋅ g ] = d d x [ f ] ⋅ g + d d x [ g ] ⋅ f {\displaystyle {\frac {d}{dx}}[f\cdot {g}]={\frac {d}{dx}}[f]\cdot {g}+{\frac {d}{dx}}[g]\cdot {f}}
Q u o t i e n t R u l e {\displaystyle \color {Red}Quotient\,Rule}
d d x [ f g ] = d d x [ f ] ⋅ g − d d x [ g ] ⋅ f g 2 {\displaystyle {\frac {d}{dx}}[{\frac {f}{g}}]={\frac {{\frac {d}{dx}}[f]\cdot {g}-{\frac {d}{dx}}[g]\cdot {f}}{g^{2}}}}
E x a m p l e s {\displaystyle \mathbf {Examples} }
E x .1 {\displaystyle \mathbf {Ex.1} }
If f ( x ) = x ⋅ e x {\displaystyle f(x)=x\cdot {e^{x}}}
f ′ ( x ) = 1 ⋅ e x + x ⋅ e x {\displaystyle f^{\prime }(x)=1\cdot {e^{x}}+x\cdot {e^{x}}} f ′ ( x ) = e x ( 1 + x ) {\displaystyle f^{\prime }(x)=e^{x}(1+x)} E x .2 {\displaystyle \mathbf {Ex.2} }
If f ( t ) = t ( a + b t ) {\displaystyle f(t)={\sqrt {t}}(a+bt)}
f ′ ( t ) = 1 2 t ( a + b t ) + t t ( b ) {\displaystyle f^{\prime }(t)={\frac {1}{2{\sqrt {t}}}}(a+bt)+t{\sqrt {t}}(b)}
E x .3 {\displaystyle \mathbf {Ex.3} }
If f ( x ) = x ⋅ g ( x ) {\displaystyle f(x)={\sqrt {x}}\cdot {g(x)}}
g ( 4 ) = 2 {\displaystyle g(4)=2}
g ′ ( 4 ) = 3 {\displaystyle g^{\prime }(4)=3}
f ′ ( x ) = 1 2 x ⋅ g ( x ) + x ⋅ g ′ ( x ) {\displaystyle f^{\prime }(x)={\frac {1}{2{\sqrt {x}}}}\cdot {g(x)}+{\sqrt {x}}\cdot {g^{\prime }(x)}} f ′ ( x ) = 1 4 ⋅ 2 + 2 ⋅ 3 {\displaystyle f^{\prime }(x)={\frac {1}{4}}\cdot {2}+{2}\cdot {3}} f ′ ( x ) = 6.5 {\displaystyle f^{\prime }(x)=6.5}
E x .4 {\displaystyle \mathbf {Ex.4} } If y = x 2 + x − 2 x 3 + 6 {\displaystyle y={\frac {x^{2}+x-2}{x^{3}+6}}} y ′ = ( x 3 + 6 ) ( 2 x + 1 ) − ( x 2 + x − 2 ) ( 3 x 2 ) ( x 3 + 6 ) 2 {\displaystyle y^{\prime }={\frac {(x^{3}+6)(2x+1)-(x^{2}+x-2)(3x^{2})}{(x^{3}+6)^{2}}}} y ′ = ( 2 x 4 + x 3 + 12 x + 6 ) − ( 3 x 4 + 3 x 3 − 6 x ) ( x 3 + 6 ) 2 {\displaystyle y^{\prime }={\frac {(2x^{4}+x^{3}+12x+6)-(3x^{4}+3x^{3}-6x)}{(x^{3}+6)^{2}}}} y ′ = − x 4 − 2 x 3 + 6 x + 12 x + 6 ( x 3 + 6 ) 2 {\displaystyle y^{\prime }={\frac {-x^{4}-2x^{3}+6x+12x+6}{(x^{3}+6)^{2}}}}
![{\displaystyle {\frac {d}{dx}}[c]=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/834381b512bc1d1fc47c2d26ce38ae71daaa1053)
![{\displaystyle {\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e100ac3ba73695d9c36ad8ada81efba4c2a01129)
![{\displaystyle {\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/27fc302c62ab11c846d1255170f40351d3461dfe)
![{\displaystyle {\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/63daf7259a950f7e09db6f16b13e760574c26bf7)
![{\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/783a58bf19ce36ca7781d7d31d78161a55406bff)
![{\displaystyle {\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/02df667a9de22b7a499cb1b37b3df322b3bfc5a4)
![{\displaystyle {\frac {d}{dx}}[f\cdot {g}]={\frac {d}{dx}}[f]\cdot {g}+{\frac {d}{dx}}[g]\cdot {f}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/57ee97cfd2a83cef00e5454484f44a5c99a73190)
![{\displaystyle {\frac {d}{dx}}[{\frac {f}{g}}]={\frac {{\frac {d}{dx}}[f]\cdot {g}-{\frac {d}{dx}}[g]\cdot {f}}{g^{2}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/33117095dc4fc9bab8b77262a36eac752e1beaa5)
