3.1 {\displaystyle \mathbf {3.1} } 1. The derivative of a constant is 0 = d d x [ c ] = 0 {\displaystyle {\frac {d}{dx}}[c]=0} 2. d d x [ x n ] = n x n − 1 {\displaystyle {\frac {d}{dx}}[x^{n}]=nx^{n-1}} 3. 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)]} 4. d d x [ f ( x ) + g ( x ) ] = d d x [ f ( x ) ] + d d x [ g ( x ) ] {\displaystyle {\frac {d}{dx}}[f(x)+g(x)]={\frac {d}{dx}}[f(x)]+{\frac {d}{dx}}[g(x)]} 5. d d x [ a x ] = l n ( a ) ⋅ a x {\displaystyle {\frac {d}{dx}}[a^{x}]=ln(a)\cdot a^{x}} 6. d d x [ e x ] = e x {\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}} E x a m p l e s {\displaystyle \mathbf {\color {Blue}{Examples}} }
1) d d x [ 5 + π ] = 0 + 0 = 0 {\displaystyle {\frac {d}{dx}}[5+\pi ]=0+0=0} 2) d d x [ 3 x ] = 3 ⋅ d d x [ x ] = 3 ( 1 ) ( x ) 1 − 1 = 3 ⋅ x 0 = 3 {\displaystyle {\frac {d}{dx}}[3x]=3\cdot {\frac {d}{dx}}[x]=3(1)(x)^{1-1}=3\cdot x^{0}=3} 3) d d x [ 9 x 2 ] = 18 x {\displaystyle {\frac {d}{dx}}[9x^{2}]=18x} 4) d d x [ x 3 + x 2 + 10 ] = 3 x 2 + 2 x + 0 {\displaystyle {\frac {d}{dx}}[x^{3}+x^{2}+10]=3x^{2}+2x+0} 5) Failed to parse (syntax error): {\displaystyle {\frac{d}{dx}} [√x] = {\frac{d}{dx}} [x^{{\frac{1}{2}}] = {\frac{1}{2x}^{{\frac{1}{2}-1}} }
\sqrt{x}
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 {Blue}Power\,Rule} d d x [ x n ] = n ⋅ x n − 1 {\displaystyle {\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1} P r o d u c t R u l e {\displaystyle \color {Red}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 {Green}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 {\color {Purple}{Examples}} } E x .1 {\displaystyle \mathbf {Ex.1} } i f f ( x ) = x ⋅ e x {\displaystyle if\,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}}} E x .2 {\displaystyle \mathbf {Ex.2} } i f f ( t ) = t ( a + b t ) {\displaystyle if\,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} } i f f ( x ) = x ⋅ g ( x ) {\displaystyle if\,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)}} E x .4 {\displaystyle \mathbf {Ex.4} } y = x 2 + x − 2 x 3 + 6 {\displaystyle y={\frac {\color {Blue}{x^{2}+x-2}}{\color {Red}{x^{3}+6}}}} d d x = y ′ = ( 2 x + 1 ) ( x 3 − 6 ) − ( x 2 + x − 2 ) ( 3 x 2 ) ( x 3 + 6 ) 2 {\displaystyle {\frac {d}{dx}}=y^{\prime }={\frac {(2x+1)(x^{3}-6)-\color {Blue}{(x^{2}+x-2)}(3x^{2})}{\color {Red}{(x^{3}+6)^{2}}}}} = ( 2 x 4 + x 4 + x 3 + 12 x + 6 − [ 3 x 4 + 3 x 2 − 6 x 2 ] ( x 3 + 6 ) 2 {\displaystyle ={\frac {(2x^{4}+x^{4}+x^{3}+12x+6-[3x^{4}+3x^{2}-6x^{2}]}{(x^{3}+6)^{2}}}} = − x 4 − 2 x 3 + 6 x 2 + 12 x + 6 ( x 3 + 6 ) 2 {\displaystyle ={\frac {-x^{4}-2x^{3}+6x^{2}+12x+6}{(x^{3}+6)^{2}}}} E x .5 {\displaystyle \mathbf {Ex.5} } y = e x 1 + x 2 ( 1 , e 2 ) {\displaystyle y={\frac {e^{x}}{1+x^{2}}}\,(1,{\frac {e}{2}})\,} d d x = e x ⋅ ( 1 + x 2 ) − e x ( 2 x ) ( 1 + x 2 ) 2 {\displaystyle {\frac {d}{dx}}={\frac {e^{x}\cdot (1+x^{2})-e^{x}(2x)}{(1+x^{2})^{2}}}} d d x | x = 1 e ( 1 + 1 ) − e ′ ( 2 ) ( 1 + 1 ) 2 = 2 e − 2 e 2 2 = 0 4 = 0 {\displaystyle {\frac {d}{dx}}|_{x=1}{\frac {e(1+1)-e^{\prime }(2)}{(1+1)^{2}}}={\frac {2e-2e}{2^{2}}}={\frac {0}{4}}=0}
![{\displaystyle {\frac {d}{dx}}[c]=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/834381b512bc1d1fc47c2d26ce38ae71daaa1053)
![{\displaystyle {\frac {d}{dx}}[x^{n}]=nx^{n-1}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d3ba3abeaf2660e0d9e7591b7051715c53770112)
![{\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)+g(x)]={\frac {d}{dx}}[f(x)]+{\frac {d}{dx}}[g(x)]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fff2f7dfe33f2ad22d0a3a717c191cf78c3286d7)
![{\displaystyle {\frac {d}{dx}}[a^{x}]=ln(a)\cdot a^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3fb6fb97381b39ec5218ede903245c8670575d12)
![{\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/783a58bf19ce36ca7781d7d31d78161a55406bff)
![{\displaystyle {\frac {d}{dx}}[5+\pi ]=0+0=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dfece50fcfde546d6d76017b25135a8ec8de2fa9)
![{\displaystyle {\frac {d}{dx}}[3x]=3\cdot {\frac {d}{dx}}[x]=3(1)(x)^{1-1}=3\cdot x^{0}=3}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/12ce4dd69616ec29e87b275a573a6503c7cec018)
![{\displaystyle {\frac {d}{dx}}[9x^{2}]=18x}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f88eae1bfa924d1c2798c91554ebf3c9141d1c69)
![{\displaystyle {\frac {d}{dx}}[x^{3}+x^{2}+10]=3x^{2}+2x+0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e7df9a7716c47d3c5b968d08176e825f9c7280e6)
![{\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}}[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)
![{\displaystyle ={\frac {(2x^{4}+x^{4}+x^{3}+12x+6-[3x^{4}+3x^{2}-6x^{2}]}{(x^{3}+6)^{2}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fdb5247377f2d161c9bc989a350a4642c753c895)
