Mathematics Dictionary Dr. K. G. Shih

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Derivatives of the logarithmic family

Subjects

Read Symbol defintion Example : y' is the first derivative of y Q01 | - Derivative of y = ln(x) Q02 | - Derivative of y = arcsinh(x) Q03 | - Derivative of y = arccosh(x) Q04 | - Derivative of y = arctanh(x) Q05 | - Derivative of y = arccsch(x) Q06 | - Derivative of y = arcsech(x) Q07 | - Derivative of y = arccoth(x) Q08 | - Q09 | - Q10 | - Reference

Answers

Q01. Derivative of y = ln(x) Use laws of logarithm : ln(A)-ln(B) = ln(A/B) and ln(A^B) = B*ln(A) dy/dx = Lim[(ln(x+h) - ln(x))/h] dy/dx = Lim[(ln(x+h)/x)/h] dy/dx = Lim[(ln(1 + h/x)/(x/h/x)] dy/dx = (1/x)*Lim[(ln(1 + h/x)*(x/h)] dy/dx = (1/x)*Lim[(ln(1 + h/x)^(x/h)] Sine Lim[(1 + 1/a)^a] = 1 as h tends to zero Hence d/dx(ln(x) = 1/x Use d/dx(e^x) = e^x and ln(e^x) = x y = ln(x) and x = e^y d/dx(x) = d/dx(e^y) 1 = (d/dy(e^y))*(dy/dx) Hence dy/dx = 1/(e^y) = 1/x Go to Begin Q02. Derivative of y = arcsinh(x) Formula : y = arcsinh(x) = ln(x + Sqr(X^2 + 1)) dy/dx = d/dx(ln(x + Sqr(X^2 + 1))) dy/dx = (1/(x + Sqr(x^2 + 1))*d/dx(x + Sqr(X^2 + 1)) dy/dx = (1/(x + Sqr(x^2 + 1))*(1 + 2*x/(2*Sqr(x^2 + 1)) dy/dx = (1/(x + Sqr(x^2 + 1))*(x + Sqr(x^2 + 1))/Sqr(x^2 + 1) dy/dx = 1/Sqr(x^2 + 1) Note Compare d/dx(arcsin(x)) with d/dx(arcsinh(x)), we can see the similarity. Go to Begin Q03. Derivative of y = arccosh(x) Formula : y = arccosh(x) = ln(x + Sqr(X^2 - 1)) dy/dx = d/dx(ln(x + Sqr(X^2 - 1))) dy/dx = (1/(x + Sqr(x^2 - 1))*d/dx(x + Sqr(X^2 - 1)) dy/dx = (1/(x + Sqr(x^2 - 1))*(1 + 2*x/(2*Sqr(x^2 - 1)) dy/dx = (1/(x + Sqr(x^2 - 1))*(x + Sqr(x^2 - 1))/Sqr(x^2 - 1) dy/dx = 1/Sqr(x^2 - 1) Note Compare d/dx(arccos(x)) with d/dx(arccosh(x)), we can see the similarity. Go to Begin Q04. Derivative of y = arctanh(x) Formula : y = arctanh(x) = ln((1 + x)/(1 - x))/2 dy/dx = d/dx(ln(1 + x)/(1 - x))/2 dy/dx = ((1 - x)/(1 + x))*d/dx((1 + x)/(1 - x))/2 dy/dx = ((1 - x)/(1 + x))*((1-x)*1 - (1+x)*(-1))/((1 - x)^2)/2 dy/dx = ((1 - x)/(1 + x))*(2/((1 - x)^2)/2 dy/dx = ((1 - x)/(1 + x))/(1 - x^2) dy/dx = +1/(1 - x^2) Note Compare d/dx(arctan(x)) with d/dx(arctanh(x)), we can see the similarity. Go to Begin Q05. Derivative of y = arccsch(x) Formula : y = arccsch(x) = ln(1/x + Sqr(1/x^2 + 1)) dy/dx = d/dx(ln(1/x + Sqr(1/x^2 + 1)) dy/dx = 1/(1/x + Sqr(1/x^2 + 1))*d/dx(1/x + Sqr(1/x^2 + 1)) dy/dx = 1/(1/x + Sqr(1/x^2 + 1))*(-1/x^2 - 2*x/(2*Sqr(1/x^2 + 1))) dy/dx = 1/(1/x + Sqr(1/x^2 + 1))*(-Sqr(1/x^2 + 1) - 1/x)/((x^2)*(Sqr(1/x^2 + 1))) dy/dx = -1/((x^2)*(Sqr(1/x^2 + 1))) dy/dx = -1/(x*Sqr(x^2 + 1)) Note Compare d/dx(arccsc(x)) with d/dx(arccsch(x)), we can see the similarity. Go to Begin Q06. Derivative of y = arcsech(x) Formula : y = arcsech(x) = ln(1/x + Sqr(1/x^2 - 1)) dy/dx = d/dx(ln(1/x + Sqr(1/x^2 - 1)) dy/dx = 1/(1/x + Sqr(1/x^2 - 1))*d/dx(1/x + Sqr(1/x^2 - 1)) dy/dx = 1/(1/x + Sqr(1/x^2 - 1))*(-1/x^2 - 2*x/(2*Sqr(1/x^2 - 1))) dy/dx = 1/(1/x + Sqr(1/x^2 - 1))*(-Sqr(1/x^2 - 1) - 1/x)/((x^2)*(Sqr(1/x^2 - 1))) dy/dx = 1/((x^2)*(Sqr(1/x^2 - 1))) dy/dx = -1/(x*Sqr(1 - x^2)) Note Compare d/dx(arcsec(x)) with d/dx(arcsech(x)), we can see the similarity. Go to Begin Q07. Derivative of y = arccoth(x) Formula : y = arccoth(x) = ln((1 + x)/(1 - x))/2 dy/dx = d/dx(ln(x + 1)/(x - 1))/2 dy/dx = ((x - 1)/(x + 1))*d/dx((x + 1)/(x - 1))/2 dy/dx = ((x - 1)/(x + 1))*((x - 1)*1 - (x + 1)*(1))/((x - 1)^2)/2 dy/dx = ((x - 1)/(x + 1))*((x - 1)*1 - (x + 1)*(1))/((x - 1)^2)/2 dy/dx = ((x - 1)/(x + 1))*(-2/((x - 1)^2)/2 dy/dx = ((x - 1)/(x + 1))*(-2/((x - 1)^2)/2 dy/dx = ((1 - x)/(1 + x))*(-1)/(1 - x^2) dy/dx = -1/(1 - x^2) Note Compare d/dx(arccot(x)) with d/dx(arccoth(x)), we can see the similarity. Go to Begin Q08. Answer Go to Begin Q09. Answer Go to Begin Q10. Reference Analytic geommetry Hyperbolic function Picture Mathematics by Dr. K. G. Shih, P129 - P135 and P175 Go to Begin

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