Calculus Volume 1

Key Equations

Calculus Volume 1Key Equations

Key Equations

 Difference quotient $Q=f(x)−f(a)x−aQ=f(x)−f(a)x−a$ Difference quotient with increment $hh$ $Q=f(a+h)−f(a)a+h−a=f(a+h)−f(a)hQ=f(a+h)−f(a)a+h−a=f(a+h)−f(a)h$ Slope of tangent line $mtan=limx→af(x)−f(a)x−amtan=limx→af(x)−f(a)x−a$ $mtan=limh→0f(a+h)−f(a)hmtan=limh→0f(a+h)−f(a)h$ Derivative of $f(x)f(x)$ at $aa$ $f′(a)=limx→af(x)−f(a)x−af′(a)=limx→af(x)−f(a)x−a$ $f′(a)=limh→0f(a+h)−f(a)hf′(a)=limh→0f(a+h)−f(a)h$ Average velocity $vave=s(t)−s(a)t−avave=s(t)−s(a)t−a$ Instantaneous velocity $v(a)=s′(a)=limt→as(t)−s(a)t−av(a)=s′(a)=limt→as(t)−s(a)t−a$
 The derivative function $f′(x)=limh→0f(x+h)−f(x)hf′(x)=limh→0f(x+h)−f(x)h$
 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 $h′(x)=n(g(x))n−1g′(x)h′(x)=n(g(x))n−1g′(x)$
 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 $ddxsec−1x=1|x|(x)2−1ddxsec−1x=1|x|(x)2−1$ Derivative of inverse cosecant function $ddxcsc−1x=−1|x|(x)2−1ddxcsc−1x=−1|x|(x)2−1$
 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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