Key Equations
row: Difference quotient | Q=f(x)−f(a)x−aQ=f(x)−f(a)x−a
row: Difference quotient with increment hh | Q=f(a+h)−f(a)a+h−a=f(a+h)−f(a)hQ=f(a+h)−f(a)a+h−a=f(a+h)−f(a)h
row: Slope of tangent line | mtan=limx→af(x)−f(a)x−amtan=limx→af(x)−f(a)x−a mtan=limh→0f(a+h)−f(a)hmtan=limh→0f(a+h)−f(a)h
row: Derivative of f(x)f(x) at aa | f′(a)=limx→af(x)−f(a)x−af′(a)=limx→af(x)−f(a)x−a f′(a)=limh→0f(a+h)−f(a)hf′(a)=limh→0f(a+h)−f(a)h
row: Average velocity | vave=s(t)−s(a)t−avave=s(t)−s(a)t−a
row: Instantaneous velocity | v(a)=s′(a)=limt→as(t)−s(a)t−av(a)=s′(a)=limt→as(t)−s(a)t−a
row: The derivative function | f′(x)=limh→0f(x+h)−f(x)hf′(x)=limh→0f(x+h)−f(x)h
row: Derivative of sine function | ddx(sinx)=cosxddx(sinx)=cosx
row: Derivative of cosine function | ddx(cosx)=−sinxddx(cosx)=−sinx
row: Derivative of tangent function | ddx(tanx)=sec2xddx(tanx)=sec2x
row: Derivative of cotangent function | ddx(cotx)=−csc2xddx(cotx)=−csc2x
row: Derivative of secant function | ddx(secx)=secxtanxddx(secx)=secxtanx
row: Derivative of cosecant function | ddx(cscx)=−cscxcotxddx(cscx)=−cscxcotx
row: The chain rule | h′(x)=f′(g(x))g′(x)h′(x)=f′(g(x))g′(x)
row: The power rule for functions | h′(x)=n(g(x))n−1g′(x)h′(x)=n(g(x))n−1g′(x)
row: Inverse function theorem | (f−1)′(x)=1f′(f−1(x))(f−1)′(x)=1f′(f−1(x)) whenever f′(f−1(x))≠0f′(f−1(x))≠0 and f(x)f(x) is differentiable.
row: Power rule with rational exponents | ddx(xm/n)=mnx(m/n)−1.ddx(xm/n)=mnx(m/n)−1.
row: Derivative of inverse sine function | ddxsin−1x=11−(x)2ddxsin−1x=11−(x)2
row: Derivative of inverse cosine function | ddxcos−1x=−11−(x)2ddxcos−1x=−11−(x)2
row: Derivative of inverse tangent function | ddxtan−1x=11+(x)2ddxtan−1x=11+(x)2
row: Derivative of inverse cotangent function | ddxcot−1x=−11+(x)2ddxcot−1x=−11+(x)2
row: Derivative of inverse secant function | ddxsec−1x=1|x|(x)2−1ddxsec−1x=1|x|(x)2−1
row: Derivative of inverse cosecant function | ddxcsc−1x=−1|x|(x)2−1ddxcsc−1x=−1|x|(x)2−1
1. ਕਤਾਰ: ਕੁਦਰਤੀ ਘਾਤਕ ਫਲਨ ਦਾ ਵਿਉਤਪੰਨ | ddx(eg(x))=eg(x)g′(x)
2. ਕਤਾਰ: ਕੁਦਰਤੀ ਲਘੂਗਣਕੀ ਫਲਨ ਦਾ ਵਿਉਤਪੰਨ | ddx(lng(x))=1g(x)g′(x)
3. ਕਤਾਰ: ਆਮ ਘਾਤਕ ਫਲਨ ਦਾ ਵਿਉਤਪੰਨ | ddx(bg(x))=bg(x)g′(x)lnb
4. ਕਤਾਰ: ਆਮ ਲਘੂਗਣਕੀ ਫਲਨ ਦਾ ਵਿਉਤਪੰਨ | ddx(logbg(x))=g′(x)g(x)lnb