Counter
Mathematics Dictionary
Dr. K. G. Shih
Derivative of exponential family
Subjects

    Read Symbol defintion Example : Lim[F(x)] means limit of F(x) when x = c
  • Q01 | - Derivative of y = exp(x) = e^x
  • Q02 | - Derivative of y = sinh(x)
  • Q03 | - Derivative of y = cosh(x)
  • Q04 | - Derivative of y = tanh(x)
  • Q05 | - Derivative of y = csch(x)
  • Q06 | - Derivative of y = sech(x)
  • Q07 | - Derivative of y = coth(x)
  • Q08 | - Higher derivative of y = exp(x)
  • Q09 | - Higher derivative of y = exp(-x)
  • Q10 | - Higher derivative of y = sinh(x)
  • Q11 | - Higher derivative of y = cosh(x)
  • Q12 | - Reference
Answers


Q01. Derivative of y = exp(x)

Formula rquired
  • The law of exponent : e^(x+y) = (e^x)*(e^y)
  • Lim[(e^x - 1)/x] = 1 when x = 0
Formula
  • y = e^x and y' = e^x
Proof
  • dy/dx = Lim[(e^(x+h) - e^x)/h] as h tend to zero
  • dy/dx = Lim[(e^(x)*(e^h) - e^x)/h]
  • dy/dx = Lim[(e^(x)*(e^h - 1)/h]
  • dy/dx = (e^x)*Lim[(e^h - 1)/h]
  • dy/dx = e^x
2nd method of proof
  • Since d/dx(ln(x)) = 1/x
  • y = e^x
  • Take ln on both sides
  • ln(y) = ln(e^x)
  • Since ln(e^x) = x
  • Hence ln(y) = x
  • Take d/dx on both sides
  • d/dx(ln(y)) = d/dx(x)
  • (d/dy(ln(y))*(dy/dx) = 1
  • (1/y)*dy/dx = 1
  • dy/dx = y = e^x
Note
  • This the only function that it will not be changed when take derivative
  • The derivative of e^x is 1 as x tends to zero

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Q02. Derivative of y = sinh(x)

Formula required
  • sinh(x) = (e^x - e^(-x))/2
  • cosh(x) = (e^x + e^(-x))/2
Prove that d/dx(sinh(x)) = cosh(x)
  • d/dx(sinh(x)) = d/dx((exp(x) - exp(-x))/2)
  • d/dx(sinh(x)) = (exp(x) + exp(-x))/2
  • d/dx(sinh(x)) = cosh(x)
Note
  • Compare with d/dx(sin(x)), we can see the similarity of sinh(x) and sin(x)

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Q03. Derivative of y = cosh(x)

Formula required
  • sinh(x) = (e^x - e^(-x))/2
  • cosh(x) = (e^x + e^(-x))/2
Prove that d/dx(sinh(x)) = cosh(x)
  • d/dx(cosh(x)) = d/dx((exp(x) + exp(-x))/2)
  • d/dx(cosh(x)) = (exp(x) - exp(-x))/2
  • d/dx(cosh(x)) = sinh(x)
Note
  • Compare with d/dx(cos(x)), we can see the similarity of cosh(x) and cos(x)

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Q04. Derivative of y = tanh(x)

Formula : y' = d/dx(tanh(x)) = sech(x)^2
  • Tanh(x) = sinh(x)/cosh(x)
  • d/dx(tanh(x)) = d/dx(sinh(x)/cosh(x))
  • d/dx(tanh(x)) = (cosh(x)*cosh(x) - sinh(x)*sinh(x))/(cosh(x)^2)
  • Since cosh(x)^2 - sinh(x)^2 = 1
  • Hence d/dx(tanh(x)) = 1/(cosh(x)^2)
  • Hence d/dx(tanh(x)) = sech(x)^2
Note
  • Compare with d/dx(tan(x), we can see the similarity of tanh(x) and tan(x)

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Q05. Derivative of y = csch(x)

Formula : y' = d/dx(csch(x)) =
  • csch(x) = 1/sinh(x)
  • d/dx(csch(x)) = d/dx(1/sinh(x))
  • d/dx(csch(x)) = (-cosh(x))/(sinh(x)^2)
  • Hence d/dx(csch(x)) = -cosh(x)/(sinh(x)^2)
  • Hence d/dx(csch(x)) = -cosh(x)*csch(x)^2 = -csch(x)*coth(x)
Note
  • Compare with d/dx(sec(x), we can see the similarity of sech(x) and sec(x)

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Q06. Derivative of y = sech(x)

Formula : y' = d/dx(sech(x)) =
  • Sech(x) = 1/cosh(x)
  • d/dx(sech(x)) = d/dx(1/cosh(x))
  • d/dx(sech(x)) = (-sinh(x))/(cosh(x)^2)
  • Hence d/dx(sech(x)) = -sinh(x)/(cosh(x)^2)
  • Hence d/dx(sech(x)) = -sinh(x)*sech(x)^2 = -sech(x)*tanh(x)
Note
  • Compare with d/dx(sec(x), we can see the similarity of sech(x) and sec(x)

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Q07. Derivative of y = coth(x)

Formula : y' = d/dx(coth(x)) = csch(x)^2
  • Coth(x) = cosh(x)/sinh(x)
  • d/dx(coth(x)) = d/dx(cosh(x)/sinh(x))
  • d/dx(coth(x)) = (sinh(x)*sinh(x) - cosh(x)*cosh(x))/(sin(x)^2)
  • Since cosh(x)^2 - sinh(x)^2 = 1
  • Hence d/dx(coth(x)) = -1/(sinh(x)^2)
  • Hence d/dx(coth(x)) = -csch(x)^2
Note
  • Compare with d/dx(cot(x), we can see the similarity of coth(x) and cot(x)

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Q08. Higher derivative of y = exp(x)

  • 1st derivative of exp(x) = exp(x)
  • 1st derivative of exp(x) = exp(x)
  • 1st derivative of exp(x) = exp(x)
  • 1st derivative of exp(x) = exp(x)
  • 1st derivative of exp(x) = exp(x)
Note
  • This the only function that it will not be changed after taking derivative

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Q09. Higher derivative of y = exp(-x)

  • 1st derivative of exp(-x) = -exp(-x)
  • 1st derivative of exp(-x) = +exp(-x)
  • 1st derivative of exp(-x) = -exp(-x)
  • 1st derivative of exp(-x) = +exp(-x)
  • 1st derivative of exp(-x) = -exp(-x)

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Q10. Higher derivative of y = sinh(x)

  • 1st derivative of sinh(x) = cosh(x)
  • 1st derivative of sinh(x) = sinh(x)
  • 1st derivative of sinh(x) = cosh(x)
  • 1st derivative of sinh(x) = sinh(x)
  • 1st derivative of sinh(x) = cosh(x)
Note
  • This one reason to have a name similar as y = sin(x)

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Q11. Higher derivative of y = cosh(x)

  • 1st derivative of cosh(x) = sinh(x)
  • 1st derivative of cosh(x) = cosh(x)
  • 1st derivative of cosh(x) = sinh(x)
  • 1st derivative of cosh(x) = cosh(x)
  • 1st derivative of cosh(x) = sinh(x)
Note
  • This one reason to have a name similar as y = sin(x)

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Q12. Reference
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