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Calculus Volume 1

Key Equations

Calculus Volume 1Key Equations

Key Equations

Difference quotient Q=f(x)f(a)xaQ=f(x)f(a)xa
Difference quotient with increment hh Q=f(a+h)f(a)a+ha=f(a+h)f(a)hQ=f(a+h)f(a)a+ha=f(a+h)f(a)h
Slope of tangent line mtan=limxaf(x)f(a)xamtan=limxaf(x)f(a)xa
Derivative of f(x)f(x) at aa f(a)=limxaf(x)f(a)xaf(a)=limxaf(x)f(a)xa
Average velocity vave=s(t)s(a)tavave=s(t)s(a)ta
Instantaneous velocity v(a)=s(a)=limtas(t)s(a)tav(a)=s(a)=limtas(t)s(a)ta
The derivative function f(x)=limh0f(x+h)f(x)hf(x)=limh0f(x+h)f(x)h
Derivative of sine function ddx(sinx)=cosxddx(sinx)=cosx
Derivative of cosine function ddx(cosx)=sinxddx(cosx)=sinx
Derivative of tangent function ddx(tanx)=sec2xddx(tanx)=sec2x
Derivative of cotangent function ddx(cotx)=csc2xddx(cotx)=csc2x
Derivative of secant function ddx(secx)=secxtanxddx(secx)=secxtanx
Derivative of cosecant function ddx(cscx)=cscxcotxddx(cscx)=cscxcotx
The chain rule h(x)=f(g(x))g(x)h(x)=f(g(x))g(x)
The power rule for functions h(x)=n(g(x))n1g(x)h(x)=n(g(x))n1g(x)
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.
Power rule with rational exponents ddx(xm/n)=mnx(m/n)1.ddx(xm/n)=mnx(m/n)1.
Derivative of inverse sine function ddxsin−1x=11(x)2ddxsin−1x=11(x)2
Derivative of inverse cosine function ddxcos−1x=−11(x)2ddxcos−1x=−11(x)2
Derivative of inverse tangent function ddxtan−1x=11+(x)2ddxtan−1x=11+(x)2
Derivative of inverse cotangent function ddxcot−1x=−11+(x)2ddxcot−1x=−11+(x)2
Derivative of inverse secant function ddxsec−1x=1|x|(x)21ddxsec−1x=1|x|(x)21
Derivative of inverse cosecant function ddxcsc−1x=−1|x|(x)21ddxcsc−1x=−1|x|(x)21
Derivative of the natural exponential function ddx(eg(x))=eg(x)g(x)ddx(eg(x))=eg(x)g(x)
Derivative of the natural logarithmic function ddx(lng(x))=1g(x)g(x)ddx(lng(x))=1g(x)g(x)
Derivative of the general exponential function ddx(bg(x))=bg(x)g(x)lnbddx(bg(x))=bg(x)g(x)lnb
Derivative of the general logarithmic function ddx(logbg(x))=g(x)g(x)lnbddx(logbg(x))=g(x)g(x)lnb
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