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Find the derivative of tan(x).

Sagot :

DᴀʀᴋPᴀʀᴀᴅᴏx

[tex]\qquad\qquad\huge\underline{{\sf Answer}}♨[/tex]

Derivative of tan(x) is sec²x

[tex]\qquad \sf  \dashrightarrow \: \therefore \dfrac{d}{dx} ( \tan(x)) = { \sec}^{2} (x)[/tex]

You can check the first principle method of derivation in attachment

View image DᴀʀᴋPᴀʀᴀᴅᴏx
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[tex]\rightarrow \sf \dfrac{d}{dx} (tan(x))[/tex]

[tex]\rightarrow \sf \dfrac{d}{dx} ( \ \dfrac{sin(x)}{cos(x)} \ )[/tex]

use the quotient rule

[tex]\rightarrow \sf \dfrac{cos(x) * \dfrac{d}{dx} (sin(x)-sin(x)*\dfrac{d}{dx}(cos(x) }{cos(x)^2}[/tex]

[tex]\rightarrow \sf \dfrac{cos(x) * cos(x)-sin(x)*(-sin(x) )}{cos(x)^2}[/tex]

[tex]\rightarrow \sf \dfrac{cos(x)^2+sin(x)^2}{cos(x)^2}[/tex]

[tex]\rightarrow \sf \dfrac{1}{cos(x)^2}[/tex]

[tex]\rightarrow \sf sec(x)^2[/tex]

used formula's :

  • cos²(x) + sin²(x) = 1
  • [tex]\sf \dfrac{1}{cos^2(x) }= sec^2(x)[/tex]
  • [tex]\sf \frac{d}{dx}[/tex] cos(x) = -sin(x)
  • [tex]\sf \frac{d}{dx}[/tex] sin(x)  = cos(x)
  • tan(x) = sin(x)/cos(x)
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