Differentiation
Mathematics for Biotechnology
100 Differentiation Questions with Detailed Solutions
| Basic to Advanced Practice
This practice set contains different types of differentiation questions including first principle, product rule, quotient rule, chain rule, trigonometric functions, logarithmic functions, exponential functions, inverse trigonometric functions, implicit differentiation, parametric differentiation and higher-order derivatives.
Part 1: Differentiation from First Principle
Q1. Find the derivative of
\( f(x)=x^2+3x \) from first principles.
\( f(x)=x^2+3x \) from first principles.
View Solution
By definition,
\[ f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h} \]Now,
\[ f(x+h)=(x+h)^2+3(x+h) \] \[ =x^2+2xh+h^2+3x+3h \]Therefore,
\[ f(x+h)-f(x)=2xh+h^2+3h \] \[ f'(x) =\lim_{h\to0} \frac{2xh+h^2+3h}{h} \] \[ =\lim_{h\to0}(2x+h+3) \]Answer:
\[ \boxed{f'(x)=2x+3} \]
Q2. Find the derivative of
\( f(x)=x^3 \) from first principles.
\( f(x)=x^3 \) from first principles.
View Solution
\[ f'(x)= \lim_{h\to0} \frac{(x+h)^3-x^3}{h} \]Using
\[ (x+h)^3=x^3+3x^2h+3xh^2+h^3 \]we get
\[ f'(x) = \lim_{h\to0} (3x^2+3xh+h^2) \]Answer:
\[ \boxed{f'(x)=3x^2} \]
Q3. Differentiate from first principles:
\[
f(x)=\frac{1}{x}
\]
View Solution
\[ f'(x)= \lim_{h\to0} \frac{\frac1{x+h}-\frac1x}{h} \] \[ = \lim_{h\to0} \frac{x-(x+h)} {hx(x+h)} \] \[ = -\lim_{h\to0} \frac1{x(x+h)} \]Answer:
\[ \boxed{f'(x)=-\frac1{x^2}} \]
Q4. Differentiate from first principles:
\[
f(x)=\sqrt{x}
\]
View Solution
\[ f'(x)= \lim_{h\to0} \frac{\sqrt{x+h}-\sqrt{x}}{h} \]Rationalizing,
\[ = \lim_{h\to0} \frac1{\sqrt{x+h}+\sqrt{x}} \]Hence,
\[ \boxed{f'(x)=\frac1{2\sqrt{x}}} \]Part 2: Algebraic Functions
Q5. Differentiate
\[
y=5x^4-3x^3+7x-9
\]
View Solution
\[ \frac{dy}{dx} =20x^3-9x^2+7 \]Answer:
\[ \boxed{20x^3-9x^2+7} \]
Q6. Differentiate
\[
y=4x^{5/2}-3x^{1/2}+2
\]
View Solution
\[ \frac{dy}{dx} = 4\left(\frac52\right)x^{3/2} - 3\left(\frac12\right)x^{-1/2} \] \[ \boxed{ \frac{dy}{dx} = 10x^{3/2}-\frac{3}{2\sqrt{x}} } \]
Q7. Differentiate
\[
y=\frac3{x^2}+\frac5{x^3}
\]
View Solution
\[ y=3x^{-2}+5x^{-3} \] \[ y'=-6x^{-3}-15x^{-4} \] \[ \boxed{ y'=-\frac6{x^3}-\frac{15}{x^4} } \]
Q8. Differentiate
\[
y=x^{2/3}+x^{3/2}
\]
View Solution
\[ y'= \frac23x^{-1/3} + \frac32x^{1/2} \] \[ \boxed{ y'= \frac{2}{3\sqrt[3]{x}} +\frac32\sqrt{x} } \]Part 3: Product Rule
Use
\[ \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx} \]
Q9. Differentiate
\[
y=x^2\sin x
\]
View Solution
\[ y'=2x\sin x+x^2\cos x \] \[ \boxed{y'=2x\sin x+x^2\cos x} \]
Q10. Differentiate
\[
y=(x^2+1)(x^3-2x)
\]
View Solution
\[ y'=(x^2+1)(3x^2-2)+(x^3-2x)(2x) \] \[ \boxed{y'=5x^4-3x^2-2} \]
Q11. Differentiate
\[
y=xe^x
\]
View Solution
\[ y'=e^x+xe^x \] \[ \boxed{y'=e^x(x+1)} \]
Q12. Differentiate
\[
y=x^2\ln x
\]
View Solution
\[ y'=2x\ln x+x \] \[ \boxed{y'=x+2x\ln x} \]Part 4: Quotient Rule
\[ \left(\frac uv\right)' = \frac{vu'-uv'}{v^2} \]
Q13. Differentiate
\[
y=\frac{x^2+1}{x}
\]
View Solution
\[ y' = \frac{2x^2-(x^2+1)}{x^2} \] \[ \boxed{ y'=1-\frac1{x^2} } \]
Q14. Differentiate
\[
y=\frac{x^2-3}{x+1}
\]
View Solution
\[ y' = \frac{2x(x+1)-(x^2-3)} {(x+1)^2} \] \[ \boxed{ y'=\frac{x^2+2x+3}{(x+1)^2} } \]
Q15. Differentiate
\[
y=\frac{\sin x}{x}
\]
View Solution
\[ \boxed{ y'= \frac{x\cos x-\sin x}{x^2} } \]
Q16. Differentiate
\[
y=\frac{e^x}{x^2}
\]
View Solution
\[ y' = \frac{x^2e^x-2xe^x}{x^4} \] \[ \boxed{ y'=\frac{e^x(x-2)}{x^3} } \]Part 5: Chain Rule
Q17. Differentiate
\[
y=(3x+2)^5
\]
View Solution
\[ y'=5(3x+2)^4(3) \] \[ \boxed{y'=15(3x+2)^4} \]
Q18. Differentiate
\[
y=(x^2+4x+1)^6
\]
View Solution
\[ y'=6(x^2+4x+1)^5(2x+4) \] \[ \boxed{ y'=12(x+2)(x^2+4x+1)^5 } \]
Q19. Differentiate
\[
y=\sqrt{1+x^2}
\]
View Solution
\[ y=(1+x^2)^{1/2} \] \[ y'= \frac12(1+x^2)^{-1/2}(2x) \] \[ \boxed{ y'=\frac{x}{\sqrt{1+x^2}} } \]
Q20. Differentiate
\[
y=(2x-1)^{-4}
\]
View Solution
\[ y'=-4(2x-1)^{-5}(2) \] \[ \boxed{ y'=-\frac8{(2x-1)^5} } \]Part 6: Trigonometric Differentiation
Q21. Differentiate
\[
y=3\sin x-4\cos x
\]
View Solution
\[ \boxed{ y'=3\cos x+4\sin x } \]
Q22. Differentiate
\[
y=\tan x+\cot x
\]
View Solution
\[ \boxed{ y'=\sec^2x-\csc^2x } \]
Q23. Differentiate
\[
y=\sin(3x+2)
\]
View Solution
\[ \boxed{ y'=3\cos(3x+2) } \]
Q24. Differentiate
\[
y=\cos(x^2)
\]
View Solution
\[ \boxed{ y'=-2x\sin(x^2) } \]
Q25. Differentiate
\[
y=\sin^2x
\]
View Solution
\[ y'=2\sin x\cos x \] \[ \boxed{y'=\sin2x} \]Part 7: Exponential and Logarithmic Functions
Q26. Differentiate
\[
y=e^{3x+1} \]
View Solution
\[ \boxed{ y'=3e^{3x+1}} \]
Q27. Differentiate
\[
y=e^{3x}
\]
View Solution
\[ \boxed{y'=3e^{3x}} \]
Q28. Differentiate
\[
y=e^{x^2+2x}
\]
View Solution
\[ \boxed{ y'=(2x+2)e^{x^2+2x} } \]
Q29. Differentiate
\[
y=2^x
\]
View Solution
\[ \boxed{ y'=2^x\ln2 } \]
Q30. Differentiate
\[
y=5^{x^2}
\]
View Solution
\[ \boxed{ y'=2x\,5^{x^2}\ln5 } \]Part 8: Logarithmic Differentiation
Q31. Differentiate
\[
y=\ln(x^2+1)
\]
View Solution
\[ \boxed{ y'=\frac{2x}{x^2+1} } \]
Q32. Differentiate
\[
y=\ln(\sin x)
\]
View Solution
\[ y'=\frac{\cos x}{\sin x} \] \[ \boxed{y'=\cot x} \]
Q33. Differentiate
\[
y=\log_a x
\]
View Solution
\[ \boxed{ y'=\frac1{x\ln a} } \]
Q34. Differentiate
\[
y=\ln\left(\frac{x+1}{x-1}\right)
\]
View Solution
\[ y=\ln(x+1)-\ln(x-1) \] \[ y'=\frac1{x+1}-\frac1{x-1} \] \[ \boxed{ y'=-\frac2{x^2-1} } \]
Q35. Differentiate
\[
y=x^x
\]
View Solution
Take logarithm:
\[ \ln y=x\ln x \] \[ \frac{y'}y=\ln x+1 \]Therefore,
\[ \boxed{ y'=x^x(1+\ln x) } \]
Q36. Differentiate
\[
y=(x^2+1)^x
\]
View Solution
\[ \ln y=x\ln(x^2+1) \] \[ \frac{y'}y = \ln(x^2+1) +\frac{2x^2}{x^2+1} \] \[ \boxed{ y'=(x^2+1)^x \left[ \ln(x^2+1) +\frac{2x^2}{x^2+1} \right] } \]
Q37. Differentiate
\[
y=x^{\sin x}
\]
View Solution
\[ \ln y=\sin x\ln x \] \[ \frac{y'}y = \cos x\ln x+\frac{\sin x}{x} \] \[ \boxed{ y'=x^{\sin x} \left( \cos x\ln x+\frac{\sin x}{x} \right) } \]Part 9: Inverse Trigonometric Functions
Q38. Differentiate
\[
y=\sin^{-1}x
\]
View Solution
\[ \boxed{ y'=\frac1{\sqrt{1-x^2}} } \]
Q39. Differentiate
\[
y=\tan^{-1}(2x)
\]
View Solution
\[ \boxed{ y'=\frac2{1+4x^2} } \]
Q40. Differentiate
\[
y=\sin^{-1}(x^2)
\]
View Solution
\[ \boxed{ y'=\frac{2x}{\sqrt{1-x^4}} } \]
Q41. Differentiate
\[
y=\cos^{-1}(3x)
\]
View Solution
\[ \boxed{ y'=-\frac3{\sqrt{1-9x^2}} } \]Part 10: Implicit Differentiation
Q42. If
\[
x^2+y^2=25,
\]
find \(dy/dx\).
View Solution
\[ 2x+2y\frac{dy}{dx}=0 \] \[ \boxed{ \frac{dy}{dx}=-\frac{x}{y} } \]
Q43. If
\[
x^3+y^3=3axy,
\]
find \(dy/dx\).
View Solution
\[ 3x^2+3y^2y'=3a(xy'+y) \] \[ x^2+y^2y'=axy'+ay \] \[ y'(y^2-ax)=ay-x^2 \] \[ \boxed{ y'=\frac{ay-x^2}{y^2-ax} } \]
Q44. If
\[
xy+y^2=x^3,
\]
find \(dy/dx\).
View Solution
\[ xy'+y+2yy'=3x^2 \] \[ (x+2y)y'=3x^2-y \] \[ \boxed{ y'=\frac{3x^2-y}{x+2y} } \]Part 11: Parametric and Higher Derivatives
Q45. If
\[
x=t^2,\qquad y=t^3,
\]
find \(dy/dx\).
View Solution
\[ \frac{dx}{dt}=2t, \qquad \frac{dy}{dt}=3t^2 \] \[ \boxed{ \frac{dy}{dx}=\frac{3t}{2} } \]
Q46. If
\[
x=a\cos t,\qquad y=a\sin t,
\]
find \(dy/dx\).
View Solution
\[ \frac{dx}{dt}=-a\sin t \] \[ \frac{dy}{dt}=a\cos t \] \[ \boxed{ \frac{dy}{dx}=-\cot t } \]
Q47. Find the second derivative if
\[
y=x^4-3x^2+5x
\]
View Solution
\[ y'=4x^3-6x+5 \] \[ \boxed{ y''=12x^2-6 } \]
Q48. Find the second derivative of
\[
y=e^x\sin x
\]
View Solution
\[ y'=e^x(\sin x+\cos x) \] \[ y'' = e^x(\sin x+\cos x) + e^x(\cos x-\sin x) \] \[ \boxed{ y''=2e^x\cos x } \]
Q49. Differentiate
\[
y=\frac{x^2\sin x}{e^x}
\]
View Solution
\[ y=x^2\sin x\,e^{-x} \] \[ y' = e^{-x} \left[ 2x\sin x+x^2\cos x-x^2\sin x \right] \] \[ \boxed{ y'= \frac{ 2x\sin x+x^2\cos x-x^2\sin x }{e^x} } \]
Q50. Differentiate
\[
y=
\ln
\left[
\frac{(x^2+1)^3\sqrt{x-1}}
{(x+2)^4}
\right]
\]
View Solution
\[ y= 3\ln(x^2+1) +\frac12\ln(x-1) -4\ln(x+2) \] \[ \boxed{ y' = \frac{6x}{x^2+1} +\frac1{2(x-1)} -\frac4{x+2} } \]Questions 51–100: Advanced Practice
Q51. Differentiate
\[
y=(2x^3-5x+1)^4
\]
View Solution
\[ \boxed{ y'=4(6x^2-5)(2x^3-5x+1)^3 } \]
Q52. Differentiate
\[
y=\sqrt{3x^2+4x+7}
\]
View Solution
\[ \boxed{ y'=\frac{3x+2} {\sqrt{3x^2+4x+7}} } \]
Q53. Differentiate
\[
y=(x^2+1)^{-3}
\]
View Solution
\[ \boxed{ y'=-\frac{6x}{(x^2+1)^4} } \]
Q54. Differentiate
\[
y=\frac1{\sqrt{2x+5}}
\]
View Solution
\[ \boxed{ y'=-\frac1{(2x+5)^{3/2}} } \]
Q55. Differentiate
\[
y=\sqrt{\frac{x+1}{x-1}}
\]
View Solution
\[ \boxed{ y'= -\frac1{(x-1)\sqrt{x^2-1}} } \]
Q56. Differentiate
\[
y=(x^3+1)\cos x
\]
View Solution
\[ \boxed{ y'=3x^2\cos x-(x^3+1)\sin x } \]
Q57. Differentiate
\[
y=x^3e^{2x}
\]
View Solution
\[ \boxed{ y'=x^2e^{2x}(3+2x) } \]
Q58. Differentiate
\[
y=(\ln x)\sin x
\]
View Solution
\[ \boxed{ y'= \frac{\sin x}{x}+\ln x\cos x } \]
Q59. Differentiate
\[
y=\frac{x^3+2}{x^2+1}
\]
View Solution
\[ \boxed{ y'= \frac{x^4+3x^2-4x}{(x^2+1)^2} } \]
Q60. Differentiate
\[
y=\frac{x\sin x}{1+x^2}
\]
View Solution
\[ \boxed{ y'= \frac{ (1-x^2)\sin x+ x(1+x^2)\cos x }{ (1+x^2)^2 } } \]
Q61. Differentiate
\[
y=\sin(2x^2+3x)
\]
View Solution
\[ \boxed{ y'=(4x+3)\cos(2x^2+3x) } \]
Q62. Differentiate
\[
y=\cos^3x
\]
View Solution
\[ \boxed{ y'=-3\cos^2x\sin x } \]
Q63. Differentiate
\[
y=\sec(3x)
\]
View Solution
\[ \boxed{ y'=3\sec(3x)\tan(3x) } \]
Q64. Differentiate
\[
y=\cot(x^2)
\]
View Solution
\[ \boxed{ y'=-2x\csc^2(x^2) } \]
Q65. Differentiate
\[
y=
\frac{\sin x+\cos x}
{\sin x-\cos x}
\]
View Solution
\[ \boxed{ y'= -\frac2{(\sin x-\cos x)^2} } \]
Q66. Differentiate
\[
y=\frac{1-\cos x}{1+\cos x}
\]
View Solution
\[ \boxed{ y'= \frac{2\sin x}{(1+\cos x)^2} } \]
Q67. Differentiate
\[
y=\sin x\cos x
\]
View Solution
\[ y'=\cos^2x-\sin^2x \] \[ \boxed{y'=\cos2x} \]
Q68. Differentiate
\[
y=\frac{\tan x}{1+\tan^2x}
\]
View Solution
\[ y=\sin x\cos x \] \[ \boxed{y'=\cos2x} \]
Q69. Differentiate
\[
y=\sin^2x+\cos^2x
\]
View Solution
\[ \sin^2x+\cos^2x=1 \] \[ \boxed{y'=0} \]
Q70. Differentiate
\[
y=\sec x+\tan x
\]
View Solution
\[ \boxed{ y'=\sec x(\tan x+\sec x) } \]
Q71. Differentiate
\[
y=e^{\sin x}
\]
View Solution
\[ \boxed{ y'=e^{\sin x}\cos x } \]
Q72. Differentiate
\[
y=e^{x^3-4x}
\]
View Solution
\[ \boxed{ y'=(3x^2-4)e^{x^3-4x} } \]
Q73. Differentiate
\[
y=\ln\sqrt{x^2+1}
\]
View Solution
\[ y=\frac12\ln(x^2+1) \] \[ \boxed{ y'=\frac{x}{x^2+1} } \]
Q74. Differentiate
\[
y=\ln(x+\sqrt{x^2+1})
\]
View Solution
\[ \boxed{ y'=\frac1{\sqrt{x^2+1}} } \]
Q75. Differentiate
\[
y=\log_{10}(x^2+1)
\]
View Solution
\[ \boxed{ y'= \frac{2x}{(x^2+1)\ln10} } \]
Q76. Differentiate
\[
y=x^{x^2}
\]
View Solution
\[ \ln y=x^2\ln x \] \[ \frac{y'}y=2x\ln x+x \] \[ \boxed{ y'=x^{x^2}(2x\ln x+x) } \]
Q77. Differentiate
\[
y=(\sin x)^x
\]
View Solution
\[ \ln y=x\ln(\sin x) \] \[ \boxed{ y' = (\sin x)^x [\ln(\sin x)+x\cot x] } \]
Q78. Differentiate
\[
y=x^{1/x}
\]
View Solution
\[ \ln y=\frac{\ln x}{x} \] \[ \boxed{ y'=x^{1/x}\frac{1-\ln x}{x^2} } \]
Q79. Differentiate
\[
y=(x+1)^{x+2}
\]
View Solution
\[ \boxed{ y' = (x+1)^{x+2} \left[ \ln(x+1)+\frac{x+2}{x+1} \right] } \]
Q80. Differentiate
\[
y=(x^2+3)^{\sin x}
\]
View Solution
\[ \boxed{ y' = (x^2+3)^{\sin x} \left[ \cos x\ln(x^2+3) + \frac{2x\sin x}{x^2+3} \right] } \]
Q81. Differentiate
\[
y=\tan^{-1}(x^2)
\]
View Solution
\[ \boxed{ y'=\frac{2x}{1+x^4} } \]
Q82. Differentiate
\[
y=\sin^{-1}(2x)
\]
View Solution
\[ \boxed{ y'=\frac2{\sqrt{1-4x^2}} } \]
Q83. Differentiate
\[
y=\cos^{-1}(x^3)
\]
View Solution
\[ \boxed{ y'=-\frac{3x^2}{\sqrt{1-x^6}} } \]
Q84. Differentiate
\[
y=\tan^{-1}\left(\frac1x\right)
\]
View Solution
\[ \boxed{ y'=-\frac1{x^2+1} } \]
Q85. Differentiate
\[
y=
\sin^{-1}
\left(
\frac{x}{\sqrt{1+x^2}}
\right)
\]
View Solution
\[ \boxed{ y'=\frac1{1+x^2} } \]
Q86. If
\[
x^2+xy+y^2=7,
\]
find \(dy/dx\).
View Solution
\[ 2x+xy'+y+2yy'=0 \] \[ \boxed{ y'=-\frac{2x+y}{x+2y} } \]
Q87. If
\[
x^2y+xy^2=6,
\]
find \(dy/dx\).
View Solution
\[ 2xy+x^2y'+y^2+2xyy'=0 \] \[ \boxed{ y'= -\frac{2xy+y^2}{x^2+2xy} } \]
Q88. If
\[
\sin(x+y)=x-y,
\]
find \(dy/dx\).
View Solution
\[ \cos(x+y)(1+y')=1-y' \] \[ \boxed{ y'= \frac{1-\cos(x+y)} {1+\cos(x+y)} } \]
Q89. If
\[
e^{x+y}=xy,
\]
find \(dy/dx\).
View Solution
\[ e^{x+y}(1+y')=y+xy' \] \[ \boxed{ y'= \frac{y-e^{x+y}} {e^{x+y}-x} } \]
Q90. If
\[
x^y=y^x,
\]
find \(dy/dx\).
View Solution
\[ y\ln x=x\ln y \] \[ y'\ln x+\frac yx = \ln y+\frac{x}{y}y' \] \[ \boxed{ y' = \frac{\ln y-y/x} {\ln x-x/y} } \]
Q91. If
\[
x=t+\frac1t,
\qquad
y=t-\frac1t,
\]
find \(dy/dx\).
View Solution
\[ \frac{dx}{dt}=1-\frac1{t^2} \] \[ \frac{dy}{dt}=1+\frac1{t^2} \] \[ \boxed{ \frac{dy}{dx} = \frac{t^2+1}{t^2-1} } \]
Q92. If
\[
x=a\cos^3t,
\qquad
y=a\sin^3t,
\]
find \(dy/dx\).
View Solution
\[ \frac{dx}{dt} = -3a\cos^2t\sin t \] \[ \frac{dy}{dt} = 3a\sin^2t\cos t \] \[ \boxed{ \frac{dy}{dx}=-\tan t } \]
Q93. If
\[
x=e^t\cos t,
\qquad
y=e^t\sin t,
\]
find \(dy/dx\).
View Solution
\[ \boxed{ \frac{dy}{dx} = \frac{\sin t+\cos t} {\cos t-\sin t} } \]
Q94. Find the second derivative of
\[
y=\sin3x
\]
View Solution
\[ y'=3\cos3x \] \[ \boxed{ y''=-9\sin3x } \]
Q95. Find the second derivative of
\[
y=xe^x
\]
View Solution
\[ y'=e^x(1+x) \] \[ \boxed{ y''=e^x(x+2) } \]
Q96. Find the second derivative of
\[
y=\ln x
\]
View Solution
\[ y'=\frac1x \] \[ \boxed{ y''=-\frac1{x^2} } \]
Q97. Differentiate
\[
y=e^{x^2}\sin(x^3)
\]
View Solution
\[ y' = 2xe^{x^2}\sin(x^3) + 3x^2e^{x^2}\cos(x^3) \] \[ \boxed{ y' = e^{x^2} \left[ 2x\sin(x^3)+3x^2\cos(x^3) \right] } \]
Q98. Differentiate
\[
y=\frac{\ln x}{x}
\]
View Solution
\[ \boxed{ y'=\frac{1-\ln x}{x^2} } \]
Q99. Differentiate
\[
y=
\tan^{-1}
\left(
\frac{2x}{1-x^2}
\right)
\]
View Solution
Let
\[ u=\frac{2x}{1-x^2} \]Then
\[ u' = \frac{2(1+x^2)} {(1-x^2)^2} \]Also,
\[ 1+u^2 = \frac{(1+x^2)^2} {(1-x^2)^2} \]Therefore,
\[ \boxed{ y'=\frac2{1+x^2} } \]
Q100. Differentiate
\[
y=
\left(
\frac{x-1}{x+1}
\right)^x
\]
View Detailed Solution
Take logarithm on both sides:
\[ \ln y = x\ln \left( \frac{x-1}{x+1} \right) \]Using logarithm properties,
\[ \ln y = x[\ln(x-1)-\ln(x+1)] \]Differentiating,
\[ \frac{y'}y = \ln \left( \frac{x-1}{x+1} \right) + x \left[ \frac1{x-1} - \frac1{x+1} \right] \]Now,
\[ \frac1{x-1}-\frac1{x+1} = \frac2{x^2-1} \]Therefore,
\[ \boxed{ y' = \left( \frac{x-1}{x+1} \right)^x \left[ \ln \left( \frac{x-1}{x+1} \right) + \frac{2x}{x^2-1} \right] } \]Important Differentiation Formulae
\[ \frac{d}{dx}(x^n)=nx^{n-1} \] \[ \frac{d}{dx}(\sin x)=\cos x \] \[ \frac{d}{dx}(\cos x)=-\sin x \] \[ \frac{d}{dx}(\tan x)=\sec^2x \] \[ \frac{d}{dx}(\cot x)=-\csc^2x \] \[ \frac{d}{dx}(\sec x)=\sec x\tan x \] \[ \frac{d}{dx}(e^x)=e^x \] \[ \frac{d}{dx}(\ln x)=\frac1x \] \[ \frac{d}{dx}(\sin^{-1}x) = \frac1{\sqrt{1-x^2}} \] \[ \frac{d}{dx}(\cos^{-1}x) = -\frac1{\sqrt{1-x^2}} \] \[ \frac{d}{dx}(\tan^{-1}x) = \frac1{1+x^2} \]Product Rule
\[ (uv)'=u'v+uv' \]Quotient Rule
\[ \left(\frac uv\right)' = \frac{vu'-uv'}{v^2} \]Chain Rule
\[ \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \]Practice Mathematics Daily — Differentiation from Basic to Advanced
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