Calculus Volume 1

# Key Equations

Calculus Volume 1Key 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
$ddx(logbg(x))=g′(x)g(x)lnbddx(logbg(x))=g′(x)g(x)lnb$