## Derivative examples

Last update: 09 July 2012

## Examples

Example: Explain why $\frac{d\mathrm{ln}x}{dx}=\frac{1}{x}.$

Since
So ${e}^{\mathrm{ln}x}\frac{d\mathrm{ln}x}{dx}=1.$ So $x\frac{d\mathrm{ln}x}{dx}=1.$ So $\frac{d\mathrm{ln}x}{dx}=\frac{1}{x}.$

Example: Find $\frac{d{\mathrm{sin}}^{-1}x}{dx}.$

Since
So $\mathrm{cos}\left({\mathrm{sin}}^{-1}x\right)\frac{d{\mathrm{sin}}^{-1}x}{dx}=1.$ So $\frac{d{\mathrm{sin}}^{-1}x}{dx}=\frac{1}{\mathrm{cos}\left({\mathrm{sin}}^{-1}x\right)}.$
So we would like to "simplify" $\mathrm{cos}\left({\mathrm{sin}}^{-1}x\right).$
Since
So $1-{\left(\mathrm{cos}\left({\mathrm{sin}}^{-1}x\right)\right)}^{2}={x}^{2}.$ So $1-{x}^{2}={\left(\mathrm{cos}\left({\mathrm{sin}}^{-1}x\right)\right)}^{2}.$
So $\mathrm{cos}\left({\mathrm{sin}}^{-1}x\right)=\sqrt{1-{x}^{2}}.$ So $\frac{d{\mathrm{sin}}^{-1}x}{dx}=\frac{1}{\mathrm{cos}\left({\mathrm{sin}}^{-1}x\right)}=\frac{1}{\sqrt{1-{x}^{2}}}.$

Example: Find $\frac{d{\mathrm{cos}}^{-1}x}{dx}.$

Since
So $-\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)\frac{d{\mathrm{cos}}^{-1}x}{dx}=1.$ So $\frac{d{\mathrm{cos}}^{-1}x}{dx}=\frac{-1}{\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)}.$
So we would like to "simplify" $\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right).$
Since
So $1-{\left(\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)\right)}^{2}={x}^{2}.$ So $1-{x}^{2}={\left(\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)\right)}^{2}.$
So $\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)=\sqrt{1-{x}^{2}}.$ So $\frac{d{\mathrm{cos}}^{-1}x}{dx}=\frac{-1}{\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)}=\frac{-1}{\sqrt{1-{x}^{2}}}.$

Example: Find $\frac{d{\mathrm{tan}}^{-1}x}{dx}.$

Since
So ${\mathrm{sec}}^{2}\left({\mathrm{tan}}^{-1}x\right)\frac{d{\mathrm{tan}}^{-1}x}{dx}=1.$ So $\frac{d{\mathrm{tan}}^{-1}x}{dx}=\frac{1}{{\mathrm{sec}}^{2}\left({\mathrm{tan}}^{-1}x\right)}.$
So we would like to "simplify" ${\mathrm{sec}}^{2}\left({\mathrm{tan}}^{-1}x\right).$
Since ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1,$ $sin2x cos2x + cos2x cos2x = 1 cos2x .$ So ${\mathrm{tan}}^{2}x+1={\mathrm{sec}}^{2}x.$
So ${\mathrm{sec}}^{2}\left({\mathrm{tan}}^{-1}x\right)={\mathrm{tan}}^{2}\left({\mathrm{tan}}^{-1}x\right)+1={\left(\mathrm{tan}\left({\mathrm{tan}}^{-1}x\right)\right)}^{2}+1={x}^{2}+1.$
So $\frac{d{\mathrm{tan}}^{-1}x}{dx}=\frac{1}{{x}^{2}+1}.$

Example: Find $\frac{d{\mathrm{cot}}^{-1}x}{dx}.$

Since
So $-{\mathrm{csc}}^{2}\left({\mathrm{cot}}^{-1}x\right)\frac{d{\mathrm{cot}}^{-1}x}{dx}=1.$ So $\frac{d{\mathrm{cot}}^{-1}x}{dx}=\frac{-1}{{\mathrm{csc}}^{2}\left({\mathrm{cot}}^{-1}x\right)}.$
So we would like to "simplify" ${\mathrm{csc}}^{2}\left({\mathrm{cot}}^{-1}x\right).$
Since ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1,$ $sin2x sin2x + cos2x sin2x = 1 sin2x .$ So $1+{\mathrm{cot}}^{2}x={\mathrm{csc}}^{2}x.$
So ${\mathrm{csc}}^{2}\left({\mathrm{cot}}^{-1}x\right)=1+{\mathrm{cot}}^{2}\left({\mathrm{cot}}^{-1}x\right)=1+{\left(\mathrm{cot}\left({\mathrm{cot}}^{-1}x\right)\right)}^{2}=1+{x}^{2}.$
So $\frac{d{\mathrm{cot}}^{-1}x}{dx}=\frac{-1}{1+{x}^{2}}.$

Example: Find $\frac{d{\mathrm{sec}}^{-1}x}{dx}.$

Since
So $\mathrm{tan}\left({\mathrm{sec}}^{-1}x\right)\mathrm{sec}\left({\mathrm{sec}}^{-1}x\right)\frac{d{\mathrm{sec}}^{-1}x}{dx}=1.$ So $\mathrm{tan}\left({\mathrm{sec}}^{-1}x\right)\cdot x\cdot \frac{d{\mathrm{sec}}^{-1}x}{dx}=1.$
So $\frac{d{\mathrm{sec}}^{-1}x}{dx}=\frac{1}{x\mathrm{tan}\left({\mathrm{sec}}^{-1}x\right)}.$
So we wouldlike to "simplify" $\mathrm{tan}\left({\mathrm{sec}}^{-1}x\right).$
Since ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1,$ $sin2x cos2x + cos2x cos2x = 1 cos2x .$ So ${\mathrm{tan}}^{2}x+1={\mathrm{sec}}^{2}x.$
${\mathrm{tan}}^{2}\left({\mathrm{sec}}^{-1}x\right)+1={\mathrm{sec}}^{2}\left({\mathrm{sec}}^{-1}x\right).$ So ${\left(\mathrm{tan}\left({\mathrm{sec}}^{-1}x\right)\right)}^{2}+1={\left(\mathrm{sec}\left({\mathrm{sec}}^{-1}x\right)\right)}^{2}.$
So ${\left(\mathrm{tan}\left({\mathrm{sec}}^{-1}x\right)\right)}^{2}+1={x}^{2}.$ So $\mathrm{tan}\left({\mathrm{sec}}^{-1}x\right)=\sqrt{{x}^{2}-1}.$
So $\frac{d{\mathrm{sec}}^{-1}x}{dx}=\frac{1}{x\sqrt{{x}^{2}-1}}.$

Example: Find $\frac{d{\mathrm{csc}}^{-1}x}{dx}.$

Since
So $-\mathrm{csc}\left({\mathrm{csc}}^{-1}x\right)\mathrm{cot}\left({\mathrm{csc}}^{-1}x\right)\frac{d{\mathrm{csc}}^{-1}x}{dx}=1.$ So $-x\mathrm{cot}\left({\mathrm{csc}}^{-1}x\right)\frac{d{\mathrm{csc}}^{-1}x}{dx}=1.$
So $\frac{d{\mathrm{csc}}^{-1}x}{dx}=\frac{-1}{x\mathrm{cot}\left({\mathrm{csc}}^{-1}x\right)}.$
So we would like to "simplify" $\mathrm{cot}\left({\mathrm{csc}}^{-1}x\right).$
Since ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1,$ $sin2x sin2x + cos2x sin2x = 1 sin2x .$
So $1+{\mathrm{cot}}^{2}x={\mathrm{csc}}^{2}x.$
So $1+{\mathrm{cot}}^{2}\left({\mathrm{csc}}^{-1}x\right).$ So $1+{\left(\mathrm{cot}\left({\mathrm{csc}}^{-1}x\right)\right)}^{2}={\left(\mathrm{csc}\left({\mathrm{csc}}^{-1}x\right)\right)}^{2}.$
So $1+{\left(\mathrm{cot}\left({\mathrm{csc}}^{-1}x\right)\right)}^{2}={x}^{2}.$ So $\mathrm{cot}\left({\mathrm{csc}}^{-1}x\right)=\sqrt{{x}^{2}-1}.$
So $\frac{d{\mathrm{csc}}^{-1}x}{dx}=\frac{-1}{x\sqrt{{x}^{2}-1}}.$

Example: Find $\frac{dy}{dx}$ when $y={\mathrm{log}}_{x}10.$

$xy = xlogx10 = 10.$ Take the derivative: $dxy dx = d(elnx)y dx = deylnx dx = eylnx ( y⋅1x + dy dx lnx ) = d10 dx =0.$ So ${e}^{y\mathrm{ln}x}\left(y\cdot \frac{1}{x}+\frac{dy}{dx}\mathrm{ln}x\right)=0.$
Solve for $\frac{dy}{dx}.$ $eylnx dy dx lnx = -eylnxy x .$ So $\frac{dy}{dx}=\frac{-{e}^{y\mathrm{ln}x}y}{x{e}^{y\mathrm{ln}x}\mathrm{ln}x}=\frac{-y}{x\mathrm{ln}x}=\frac{{\mathrm{log}}_{x}10}{x\mathrm{ln}x}.$

Example: Find the third derivative of ${2}^{x}$ with respect to $x.$

$y = 2x. dy dx = d2x dx = d(eln2)x dx = dexln2 dx = exln2 (ln2) = (eln2)x ln2 = 2xln2. y = d2y dx2 = d dx ( dy dx ) = d2xln2 dx = ln2⋅2xln2 = (ln2)2 2x. d3y dx3 = d dx ( d2y dx2 ) = d dx ( (ln2)2 2x ) = (ln2)2 2xln2 = (ln2)3 2x.$

Example: If $y=a\mathrm{cos}\left(\mathrm{ln}x\right)+b\mathrm{sin}\left(\mathrm{ln}x\right)$ show that ${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}+y=0.$

$dy dx = a(-sin(lnx)) 1x + bcos(lnx) 1x = -asin(lnx)x-1 + bcos(lnx) x-1, d2y dx2 = -acos(lnx)1x x-1 + (-a) sin(lnx) (-1)x-2 + (-b) sin(lnx) 1xx-1 + bcos(lnx) (-1)x-2 = -acos(lnx) + asin(lnx) - bsin(lnx) - bcos(lnx) x2 = 1x2 ( (a-b) sin(lnx) - (a+b) cos(lnx) ).$ So $LHS = x2 d2y dx2 + x dy dx + y = x21x2 ( (a-b) sin(lnx) - (a+b) cos(lnx) ) + x( -asin(lnx)x-1 + bcos(lnx) x-1 ) + acos(lnx) + bsin(lnx) = (a-b) sin(lnx) - (a+b) cos(lnx) - asin(lnx) + bcos(lnx) + bsin(lnx) + acos(lnx) = 0.$

Example: Find $\frac{dy}{dx}$ when $a\mathrm{sin}\left(xy\right)+b\mathrm{cos}\left(\frac{x}{y}\right)=0.$

Take the derivative: $0 = acos(xy) ( x dy dx + 1⋅y ) + (-b)sin( xy ) ( x(-1)y-2 dy dx + 1⋅y-1 ) = acos(xy)x dy dx + acos(xy)y + bsin( xy ) xy2 dy dx - bsin( xy ) y-1.$ Solve for $\frac{dy}{dx}.$ $acos(xy)x dy dx + bsin( xy ) xy2 dy dx = acos(xy)y - bsin( xy )y-1.$ So $dy dx = acos(xy)y - bsin( xy )y-1 acos(xy)x + bsin( xy ) xy2 = acos(xy) y3 - bsin( xy )y acos(xy) xy2 + bsin( xy )x .$

Example: Find $\frac{dy}{dx}$ when $y={\mathrm{tan}}^{-1}\left(\frac{a}{x}\right)\cdot {\mathrm{cot}}^{-1}\left(\frac{x}{a}\right).$

$dy dx = tan-1( ax ) ( -1 1+( xa )2 ) 1a + 1 1+( xa )2 (-1) ax-2 cot-1( xa ) = -tan-1( ax ) a+x2a + -cot-1( xa )a x2+a2 = -tan-1( ax )a a2+x2 + -cot-1( xa )a x2+a2 = ( -a a2+x2 ) ( tan-1( ax ) + cot-1( xa ) ).$ If $\frac{a}{x}=\mathrm{tan}z$ then $\frac{x}{a}=\mathrm{cot}z$ and $z={\mathrm{tan}}^{-1}\left(\frac{a}{x}\right)={\mathrm{cot}}^{-1}\left(\frac{x}{a}\right).$ So $dy dx = ( -a a2+x2 ) ( tan-1( ax ) + tan-1( ax ) ) = -2atan-1( ax ) a2+x2 .$

Example: Find $\frac{d{e}^{x}}{dx}.$

$dex dx = d dx ( 1+x + x2 2! + x3 3! + x4 4! + x5 5! + x6 6! + x7 7! +⋯ ) = 0+1+ 1 2! 2x + 1 3! 3x2 + 1 4! 4x3 + 1 5! 5x4 + 1 6! 6x5 + 1 7! 7x6 +⋯ = 1+122x + 1 3⋅2! 3x2 + 1 4⋅3! 4x3 + 1 5⋅4! 5x4 + 1 6⋅5! 6x5 + 1 7⋅6! 7x6 + ⋯ = 1+x + x2 2! + x3 3! + x4 4! + x5 5! + x6 6! + x7 7! +⋯ = ex.$

Example: Find $\frac{d\mathrm{sin}x}{dx}.$

$dsinx dx = d dx ( x - x3 3! + x5 5! - x7 7! + x9 9! - x11 11! + x13 13! - ⋯ ) = 1 - 1 3! 3x2 + 1 5! 5x4 - 1 7! 7x6 + 1 9! 9x8 - 1 11! 11x10 + 1 13! 13x12 - ⋯ = 1 - x2 2! + x4 4! - x6 6! + x8 8! - x10 10! + x12 12! - ⋯ = cosx.$

Example: Find $\frac{d\mathrm{cos}x}{dx}.$

$dcosx dx = d dx ( 1 - x2 2! + x4 4! - x6 6! + x8 8! - x10 10! + x12 12! - ⋯ ) = 0 - 1 2! 2x + 1 4! 4x3 - 1 6! 6x5 + 1 8! 8x7 - 1 10! 10x9 + 1 12! 12x11 - ⋯ = - x + x3 3! - x5 5! + x7 7! - x9 9! + x11 11! - ⋯ = -( x - x3 3! + x5 5! - x7 7! + x9 9! - x11 11! + ⋯ ) = -sinx.$

Example: Find $\frac{d\mathrm{tan}x}{dx}.$

$dtanx dx = d dx ( sinx cosx ) = d dx ( sinx(cosx)-1 ) = sinx d(cosx)-1 dx + dsinx dx (cosx)-1 = sinx(-1) (cosx)-2 dcosx dx + cosx⋅ 1 cosx = - sinx cos2x (-sinx) + 1 = sin2x cos2x +1 = sin2x+cos2x cos2x = 1 cos2x = sec2x.$

Example: Find $\frac{d\mathrm{sec}x}{dx}.$

$dsecx dx = d dx ( 1cosx ) = d dx ((cosx)-1) = (-1) (cosx)-2 dcosx dx = - 1 cos2x (-sinx) = sinx cos2x = sinx cosx ⋅ 1 cosx = tanxsecx.$

Example: Find $\frac{d\mathrm{csc}x}{dx}.$

$dcscx dx = d dx (1sinx) = d dx ((sinx)-1) = (-1) (sinx)-2 dsinx dx = - 1 sin2x (cosx) = - cosx sin2x = - cosx sinx ⋅ 1 sinx = -cotxcscx.$

Example: Find $\frac{d\mathrm{cot}x}{dx}.$

$dcotx dx = d dx ( cosx sinx ) = d dx ( cosx(sinx)-1 ) = cosx d(sinx)-1 dx + dcosx dx (sinx)-1 = cosx(-1) (sinx)-2 dsinx dx + (-sinx) ⋅ 1 sinx = - cosx sin2x ⋅ cosx - 1 = -cos2x sin2x - 1 = -cos2x-sin2x sin2x = -1 sin2x = csc2x.$

Example: Find $\frac{d\mathrm{sinh}x}{dx}.$

$dsinhx dx = d dx ( x + x3 3! + x5 5! + x7 7! + x9 9! + x11 11! + x13 13! + ⋯ ) = 1 + 1 3! 3x2 + 1 5! 5x4 + 1 7! 7x6 + 1 9! 9x8 + 1 11! 11x10 + 1 13! 13x12 + ⋯ = 1 + x2 2! + x4 4! + x6 6! + x8 8! + x10 10! + x12 12! + ⋯ = coshx.$

Example: Find $\frac{d\mathrm{cosh}x}{dx}.$

$dcoshx dx = d dx ( 1 + x2 2! + x4 4! + x6 6! + x8 8! + x10 10! + x12 12! + ⋯ ) = 0 + 1 2! 2x + 1 4! 4x3 + 1 6! 6x5 + 1 8! 8x7 + 1 10! 10x9 + 1 12! 12x11 + ⋯ = x + x3 3! + x5 5! + x7 7! + x9 9! + x11 11! + ⋯ = sinhx.$

Example: Find $\frac{d\mathrm{tanh}x}{dx}.$

$dtanhx dx = d dx ( sinhx coshx ) = d dx ( sinhx(coshx)-1 ) = sinhx d(coshx)-1 dx + dsinhx dx (coshx)-1 = sinhx(-1) (coshx)-2 dcoshx dx + coshx⋅ 1 coshx = - sinhx cosh2x ⋅sinhx+1 = - sinh2x cosh2x +1 = -sinh2x+cosh2x cosh2x = 1 cosh2x = sech2x.$

Example: Find $\frac{d\mathrm{sech}x}{dx}.$

$dsechx dx = d dx ( 1coshx ) = d dx ((coshx)-1) = (-1) (coshx)-2 dcoshx dx = - 1 cosh2x ⋅sinhx = - sinhx cosh2x = - sinhx coshx ⋅ 1 coshx = -tanhxsechx.$

Example: Find $\frac{d\mathrm{csch}x}{dx}.$

$dcschx dx = d dx (1sinhx) = d dx ((sinhx)-1) = (-1) (sinhx)-2 dsinhx dx = - 1 sinh2x (coshx) = - coshx sinh2x = - coshx sinhx ⋅ 1 sinhx = -cothxcschx.$

Example: Find $\frac{d\mathrm{coth}x}{dx}.$

$dcothx dx = d dx ( 1 tanhx ) = d(tanhx)-1 dx = (-1) (tanhx)-2 dtanhx dx = - 1 tanh2x dtanhx dx = - 1 tanh2x ⋅ sech2x = - 1 sinh2x cosh2x ⋅ 1 cosh2x = - 1 sinh2x = -csch2x.$

References?