Derivatives

Click on the ANSWER tab for general solutions and examples for each.

 

Basic Derivatives

1. $latex \frac\,\,$                            [expand title="ANSWER" trigclass="my_special_class"]$latex \begin\frac\,\,=(n)'\\Ex1:\,\,\frac4-15+=20-45+\frac{^{\frac}}\\Ex2:\,\,\frac7-+10{^{\frac}}=-28+3+5{^{-\frac}}\\=-\frac{}+\frac{}+\frac{{^{\frac}}}\end$
Proof [/expand]

2. $latex \frac\,\,sin\,u$                        [expand title="ANSWER" trigclass="my_special_class"]$latex \frac\,\,sin\,u=\cos \,u\,'$    
Proof [/expand]

3. $latex \frac\,\,\cos \,u$                       [expand title="ANSWER" trigclass="my_special_class"]$latex \frac\,\,\cos \,u=-\sin \,u\,'$  
Proof [/expand]

4. $latex \frac\,\,\frac$                             [expand title="ANSWER" trigclass="my_special_class"]$latex \frac\,\,\frac=-\frac{{^}}\,'$[/expand]

5. $latex \frac\,\,\tan \,u$                        [expand title="ANSWER" trigclass="my_special_class"] $latex \frac\,\,\tan \,u={{\sec }^}u\,'$  
Proof  [/expand]

6. $latex \text\,\,\text\,\,\,\,\frac\,\,f(x)\bullet g(x)\,\,or\,\,\frac\,\,uv$              [expand title="ANSWER" trigclass="my_special_class"]$latex \begin\frac\,\,f(x)\bullet g(x)=\,\,\,\,\,'(x)\bullet g(x)+f(x)\bullet '(x)\,\,\,\,\\or\,\,\\\frac\,\,uv='v+u'\\Ex1:\,\,\frac8\tan x=16x\tan x+8{{\sec }^}x\\Ex2:\,\,\frac\sin x\cos x=\cos x\cos x+\sin x(-\sin x)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\cos }^}x-{{\sin }^}x\end$ $latex \beginEx3:\,\,\frac\sqrt=2x{{(x-5)}^{\frac}}+\frac{{(x-5)}^{-\frac}}\\\,\,\,\,\,\,\,\,\,\,now\,\,factor:\,\,=x{{(x-5)}^{-\frac}}(2(x-5)+\fracx)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=x{{(x-5)}^{-\frac}}(2x-10+\fracx)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{\sqrt}(\fracx-10)\,\,\,\,\,or\,\,\,\,\frac{2\sqrt}\,\end$[/expand]

7. $latex \text\,\,\text\,\,\,\,\frac\,\,\frac\,\,or\,\,\frac\,\,\frac$                        [expand title="ANSWER" trigclass="my_special_class"]$latex \begin\ Ex1:\,\,\frac\frac{}=\frac{\cos x\,-\sin x\,(2x)}{{{()}^}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{\cos x\,-2x\sin x\,}{}=\frac{}\\Ex2:\,\,\frac500(1\,\,\,+\,\,\frac{50+{{t}^}})=\frac(500\,\,+\,\,\frac{50+{{t}^}})\\=\frac{(50+{{t}^})2000-2000t(2t)}{{{(50+{{t}^})}^}}\\\frac{100,000+2000{{t}^}-4000{{t}^}}{{{(50+{{t}^})}^}}=\frac{-2000{{t}^}+100,000}{{{(50+{{t}^})}^}}\\=\frac{-2000({{t}^}-50)}{{{(50+{{t}^})}^}}\\Ex3:\,\,\frac\,\,\,\frac=\frac{{{(1-\cos x)}^}}\\=\frac{\cos x-{{\cos }^}x-{{\sin }^}x}{{{(1-\cos x)}^}}\\=\frac{\cos x-({{\cos }^}x+{{\sin }^}x)}{{{(1-\cos x)}^}}\,\,\,\,\,\,\,\,remember\,\,\,{{\sin }^}x+{{\cos }^}x=1\\=\frac{{{(1-\cos x)}^}}=\frac{{{(1-\cos x)}^}}=\frac\end$[/expand]

8. $latex \text\,\,\text:\,\,\fracf(g(2x))$                                       [expand title="ANSWER" trigclass="my_special_class"] $latex \fracf(g(2x))='(g(2x))\bullet '(2x)\bullet 2$ 
$latex \begin\text\,\,\text\,\,\,\text\\1.\,\,h(x)=10\sin (7x-9)\\'(x)=10\cos (7x-9)\bullet 7\\=70\cos (7x-9)\\\\2.\,\,p(x)=\sqrt[3]{7+5x-3}={{(7+5x-3)}^{\frac}}\\'(x)=\frac{{(7+5x-3)}^{-\frac}}(14x+5)\\=\frac{3{{(7+5x-3)}^{\frac}}}\\\\3.\,\,k(x)=10(7x-9)\\'(x)=40{{\cos }^}(7x-9)\bullet (-\sin (7x-9)\bullet 7)\\=-280{{\cos }^}(7x-9)\sin (7x-9)\\\\4.\,\,\text\,\,\text\,\,\text\\f(x)=4{{(5x-8)}^{\frac}}\\'(x)=12{{(5x-8)}^{\frac}}+4(\frac){{(5x-8)}^{\frac}}(5)\\=12{{(5x-8)}^{\frac}}+30{{(5x-8)}^{\frac}}\\=6{{(5x-8)}^{\frac}}[2(5x-8)+5x]\\=6{{(5x-8)}^{\frac}}(15x-16)\\\\5.\,\,h(x)=10f(g(8x))\\'(x)=10'(g(8x))\bullet '(8x)\bullet 8\\=80'(g(8x))\bullet '(8x)\\\\6.\,\,\,y={{\sin }^}(4x){{\cos }^}(5x)\\'=2\sin (4x)\cos (4x)(4)\bullet {{\cos }^}(4x)+{{\sin }^}(4x)3{{\cos }^}(5x)\bullet (-\sin (5x))\bullet (5)\\=8\sin (4x)\cos (4x){{\cos }^}(4x)-15{{\sin }^}(4x){{\cos }^}(5x)\sin (5x)\\\end$ [/expand]

 

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Trigonometric Functions

1. $latex \frac\sin u$                [expandsub1 title="ANSWER" id="trig1″ trigclass="my_special_class"]$latex \begin\frac\sin u\,\,=\cos u\bullet '\\Ex:\,\,\frac\sin (7x-12)\,\,=\cos (7x-12)\bullet 7\\=7\cos (7x-12)\end$[/expandsub1]

2. $latex \frac\cos u$               [expandsub1 title="ANSWER" id="trig2″ trigclass="my_special_class"]$latex \begin\frac\cos u\,\,=-\sin u\bullet '\\Ex:\,\,\frac\cos (+2x-3)\,\,=-\sin (+2x-3)\bullet (2x+2)\\=-(2x+2)\sin (+2x-3)\end$[/expandsub1]

3. $latex \frac\tan \,u$               [expandsub1 title="ANSWER" id="trig3″ trigclass="my_special_class"]$latex \begin\frac\tan \,u\,=u\bullet '\\Ex:\,\,\frac\tan (5-7x)\,\,=(5-7x)\bullet (10x-7)\\=(10x-7)(5-7x)\end$[/expandsub1]

4. $latex \frac\sec \,u$               [expandsub1 title="ANSWER" id="trig4″ trigclass="my_special_class"]$latex \begin\frac\sec \,u\,=\sec u\tan u\bullet '\\Ex:\,\,\frac\sec (-12)\,\,=\sec (-12)\bullet \tan (-12)\bullet 2x\\=2x\sec (-12)\bullet \tan (-12)\end$[/expandsub1]

5. $latex \frac\csc \,u$               [expandsub1 title="ANSWER" id="trig5″ trigclass="my_special_class"]$latex \beginEx:\,\,\frac\csc (\sqrt-12)\,\,=-\csc (\sqrt-12)\bullet \cot (\sqrt-12)\bullet \frac{^{-\frac}}\\=-\frac{^{-\frac}}\csc (\sqrt-12)\bullet \cot (\sqrt-12)\end$[/expandsub1]

6. $latex \frac\cot \,u$               [expandsub1 title="ANSWER" id="trig6″ trigclass="my_special_class"]$latex \begin\frac\cot \,u\,=-{{\csc }^}u\bullet '\\Ex:\,\,\frac\cot (+{^{\frac}})\,\,=-{{\csc }^}(+{^{\frac}})\bullet (2x+\frac{^{-\frac}})\\=-(2x+\frac{^{-\frac}}){{\csc }^}(+{^{\frac}})\end$[/expandsub1]

 

 

Inverse Trigonometric Functions

1. $latex \frac\arcsin u$                [expandsub2 title="ANSWER" id="invtrig1″ trigclass="my_special_class"]$latex \begin\frac\arcsin u=\frac{\sqrt{1-{^}}}\,\,\,and\,\,\,\,\,\frac\arccos u=\frac{\sqrt{1-{^}}}\\Ex:\frac\arcsin (7x-1)=\frac{\sqrt{1-{{(7x-1)}^}}}\end$[/expandsub2]

2. $latex \frac\arctan u$               [expandsub2 title="ANSWER" id="invtrig2″ trigclass="my_special_class"]$latex \begin\frac\arctan u=\frac{1+{^}}\,\,\,and\,\,\,\,\,\frac\text\,u=\frac{1+{^}}\\Ex:\frac\arctan (-x)=\frac{1+{{(-x)}^}}\end$[/expandsub2]

3. $latex \frac\text\,u$               [expandsub2 title="ANSWER" id="invtrig3″ trigclass="my_special_class"]$latex \begin\frac\text\,u=\frac{\left| u \right|\sqrt{{^}-1}}\,\,\,and\,\,\,\,\,\frac\text\,u=\frac{\left| u \right|\sqrt{{^}-1}}\\Ex:\frac\text({^})=\frac{\left| {^} \right|\sqrt{{^{6}}-1}}\end$[/expandsub2]

 

Exponential Functions

1. $latex \frac{{e}^}$                [expandsub3 title="ANSWER" id="Exponential1″ trigclass="my_special_class"]$latex \begin\frac{{e}^}={{e}^}\,'\\Ex1:\,\,\frac{{e}^{4x-}}=(4-2x){{e}^{4x-}}\\Ex2:\,\,\frac=8\sin (8x)\\Ex3:\,\,\frac\arctan (3x){{e}^{-}}\\=\frac{1+9}{{e}^{-}}+\arctan (3x)\bullet (-2x){{e}^{-}}\\={{e}^{-}}\left( \frac{1+9}-2x\arctan (3x) \right)\\\end$[/expandsub3]

2. $latex \frac{^}$                [expandsub3 title="ANSWER" id="Exponential2″ trigclass="my_special_class"]$latex \begin\frac{^}=\ln a\bullet {^}\,'\\Ex1:\,\,\frac{^{{^}}}=\ln 4\bullet {^{{^}}}\bullet 3=3\ln 4\bullet {^{{^}}}\\Ex2:\,\,\frac{^{-k}}=\ln 7\bullet {^{-k}}\bullet -2kx\\=-2kx\ln 7\bullet {^{-k}}\,\,\text\,\,\text\,\,\text\,\,\text\,\,\text\\Ex3:\,\,\frac{{\sin }^}(2x){^}\\=3{{\sin }^}(2x)\cos (2x)\bullet 2\bullet {^}+{{\sin }^}(2x)\bullet \ln k\bullet {^}(-2)\\=6{{\sin }^}(2x)\cos (2x)\bullet {^}-2{{\sin }^}(2x)\ln k\bullet {^}\\=2{{\sin }^}(2x){^}(3\cos (2x)-\sin (2x)\ln k)\\\,\text\,\,\text\,\,\text\,\,\text\,\,\text\\\end$[/expandsub3]

 

Logarithmic Functions

1. $latex \frac\ln u$                [expandsub4 title="ANSWER" id="log1″ trigclass="my_special_class"]$latex \begin\frac\ln u=\frac{{'}}\\Ex1:\,\,\frac\ln (8-)=\frac{(8-)}\\Ex2:\,\,\frac\ln (\arctan (4x))=\frac{\frac{1+16}}\\=\frac{(1+16)\arctan (4x)}\\Ex3:\,\,\frac4\ln (x-1)=8x\ln (x-1)+\frac{4(1)}\\=8x\ln (x-1)+\frac\\\end$[/expandsub4]

2. $latex \frac{{\log }_}u$             [expandsub4 title="ANSWER" id="log2″ trigclass="my_special_class"]$latex \begin\frac{{\log }_}u=\text\,\,\text\,\,\text\,\,\text\,\,'\text\!\!'\!\!\text\,\,\text\,\,\text\,\,\text\\=\frac\frac=\frac{{'}}\,\,\text\,\,\ln a\,\,\text\,\,\text\,\,\text\,\,\text\\Ex1:\,\,\frac{{\log }_}(8-)=\frac\frac{\ln (8-)}\\=\frac{\ln 3(8-)}\\Ex2:\,\,\frac(-3x+12)=\frac\frac{\ln (-3x+12)}\\=\frac{\ln 8(-3x+12)}\\Ex3:\,\,\frac{{\log }_}(x-1)=\frac\frac\\=2x\frac+\frac\\=\frac\left[ 2\ln (x-1)+\frac \right]\\\end$[/expandsub4]

 

 

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