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Differential Calculus (Definition)
f'(x) = lim [f(x+▲x) -f(x)] / [▲x]
(d/dx) [cu] = cu'
(General and Logarithmic Differentiation Rules)
(General and Logarithmic Differentiation Rules)
(d/dx) [u ± v] = u' ± v'
(d/dx) [u/v] = (vu'-uv')/(v^2)
(General and Logarithmic Differentiation Rules)
(d/dx) [c] = 0
(General and Logarithmic Differentiation Rules)
(d/dx) [u^n] = n(u^(n-1))u'
(General and Logarithmic Differentiation Rules)
(d/dx) [x] = 1
(General and Logarithmic Differentiation Rules)
(d/dx) [e^u] = (e^u)u'
(General and Logarithmic Differentiation Rules)
(d/dx) [f(g(x))] = f'(g(x))g'(x)
(General and Logarithmic Differentiation Rules)
(Derivatives of the Trigonometric Functions)
(d/dx) [sin u] = (cos u)u'
(Derivatives of the Trigonometric Functions)
(d/dx) [cos u] = -(sin u)u'
(Derivatives of the Trigonometric Functions)
(d/dx) [tan u] = (sec^2 u)u'
(Derivatives of the Trigonometric Functions)
(d/dx) [sec u] = (sec u tan u)u'
(Derivatives of the Inverse Trigonometric Functions)
(d/dx) [arcsin u] = [u'] / [√1-(u^2)]
(Derivatives of the Inverse Trigonometric Functions)
(d/dx) [arccos u] = -[u'] / [√1-(u^2)]
(Derivatives of the Inverse Trigonometric Functions)
(d/dx) [arctan u] = [u'] / [1+(u^2)]
Derivatives of Inverse Functions
If y=f(x) and x=f^-1(y) are differentiable inverse functions, then their derivatives are reciprocals: (dx/dy) = 1 / (dy/dx)
Logarithmic Differentiation
It is often advantageous to use logarithms to differentiate certain functions.
Mean Value Theorem (MVT)
If f is continuous on [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that
L'Hôpital's Rule
If lim f(x)/g(x) is an indeterminate of the form 0/0 or ∞/∞, and if lim f'(x)/g'(x) exists, then lim f(x)/g(x) = lim f'(x)/g'(x)
Tangent and Normal Lines
The derivative of a function at a point is the slope of the tangent line. The normal line is the line that is perpendicular to the tangent line at the point of tangency.
Extreme Value Theorem
If a function f(x) is continuous on a closed interval, the f(x) has both a maximum and a minimum value in the interval.
Curve Sketching
The values of derivatives and higher-order derivatives can be used to approximate the shape of f. (See packet for specific situations*)
(x)∨(n+1) = [x∨n] - [f(x∨n)/f'(x∨n)]
Newton's Method for Approximating Zeroes of a Function
Optimization Problems
Calculus can be used to solve practical problems requiring maximum or minimum values. (See example problem*)
Rate-of-Change Problems
Related rates of Change: calculus can be used to find the rate of change of two or more variable that are functions of time t by differentiating with respect to t.