Calculus Cheat Sheet Derivatives

Derivatives Definition and Notation If y = f(x) then the derivative is defined to be f' * (x) = lim h -> 0 (f(x + h) - f(x))/h If y = f(x) then all of the following are equivalent notations for the derivative. f' * (pi) = y' = d/dx (f) = d/dx (y) = d/dx (f(x)) = D * f' * (x) If y = f(x) of the following are equivalent notations for derivative evaluated at x = a f^ prime (a)=y^ prime | a a = df d tau | z=a = dv d tau | z = a = D * f' * (a) Interpretation of the Derivative If y = f(x) then. 1. m=f^ prime prime (a) is the slope of the tangent line to y = f(x) at x = a_{i} the equation of the tangent line at x = a is given by y = f(a) + f' * (a)(x - a) 2. f' * (a) is the instantaneous rate of change of f(x) at x = a 3. If f(x) is the position of an object at time x then f' * (a) is the velocity of the object at x = a Basic Properties and Formulas If f(x) and g(x) are differentiable functions (the derivative exists), c and are any real numbers. 1. (cf)' = c * f' * (x) 5. d/dx (c) = 0 2. (f plus/minus g)' = f' * (x) plus/minus g' * (x) 6. d dx (x^ n )= n * x ^ (n - 1) - Power Rule 3. (fg)' = f' * g + f * g' - Product Rule 7. d/dx (f(g(x))) = f' * (g(x)) * g' * (x) (f/g)' = (f' * g - f * g')/(g ^ 2) Quotient Rule This is the Chain Rule Common Derivatives d/dx (x) = 1 d/dx (csc(x)) = - csc(x) * cot x d/dx (a ^ x) = a ^ z * ln(a) d/dx (sin x) = cos x d/dx (cot x) = - csc^2 x d/dx (e ^ x) = e ^ x d/dx (cos x) = - sin x d/dx (arcsin(x)) = 1/(sqrt(1 - x ^ 2)) d/dx (ln(x)) = 1/x, x > 0 d/dx (tan x) = sec^2 x d/dx (arccos(x)) = - 1/(sqrt(1 - x ^ 2)) d/dx (ln|x|) = 1/x ,x ne0 d/dx (sec(x)) = sec(x) * tan x d/dx (arctan(x)) = 1/(1 + x ^ 2) d/dx (log_x(x)) = 1/(x * ln(a)), x > 0 Chain Rule Variants The chain rule applied to some specific functions. 1. d/dx ([f(x)] ^ n) = n * [f(x)] ^ (n - 1) * f' * (x) 5. d/dx (cos[f(x)]) = - f' * (x) * sin[f(x)] 2. d/dx (e ^ (f(x))) = f' * (x) * e ^ (f(x)) 6. d/dx (tan[f(x)]) = f' * (x) * sec^2 [f(x)] 3. d/dx (ln[f(x)]) = (f' * (x))/(f(x)) 7. d/dx (sec[f(x)]) = f' * (x) * sec[f(x)] * tan[f' * (x)] 4. d/dx (sin[f(x)]) = f' * (x) * cos[f(x)] 8. d/dx (tau ^ - 1 * [f(x)]) = (f' * (x))/(1 + [f(x)] ^ 2) Higher Order Derivatives The Second Derivative is denoted as f^ * (x) = f ^ (2) * (x) = (d ^ 2 * f)/(d * x ^ 2) and is defined as f^ prime prime (x) = (f' * (x))' te the derivative of the first derivative, f' * (x) The n ^ \# Derivative is denoted as f ^ (n) * (x) = (d ^ n * f)/(d * x ^ n) and is defined as f ^ (n) * (x) = (f ^ (n - 1) * (x)) ^ r te the derivative of the (n - 1) ^ (nt) derivative, f ^ (n - 1) * (x) Implicit Differentiation Find y' if e ^ (2x*theta*y) + x ^ 3 * y ^ 2 = sin(y) + 11x Remember y = y(x) here, so products/quotients of x and y will use the product/quotient rule and derivatives of y will use the chain rule. The "trick" is to differentiate as normal and every time you differentiate a y you tack on a y ^ * (from the chain rule). After differentiating solve for y' epsilon ^ (2x - 9y) * (2 - 9 * y') + 3x ^ 2 * y ^ 2 + 2x ^ 3 * y * y' = cos(y) * y' + 11 aligned 2e ^ (2x - by) - 9 * y' * e ^ (2x - by) + 3x ^ 2 * y ^ 2 + 2x ^ 3 * y * y' =cos(y)y^ prime +11 Rightarrow y^ prime = 11-2e^ 2x-by -3x^ 2 y^ 2 2x^ 3 y-9e^ 2x- theta y -cos(y) aligned (2x ^ 3 * y - 9e ^ (2x - 9y) - cos(y)) * y' = 11 - 2e ^ (2x - 9y) - 3x ^ 2 * y ^ 2 Increasing/Decreasing - Concave Up/Concave Down Critical Points x = c is a critical point of f(x) provided either 1. f' * (c) = 0 or 2. f' * (c) doesn't exist. Increasing/Decreasing 1. If f' * (x) > 0 for all x in an interval / then f(x) is increasing on the interval J. 2. If f' * (x) < 0 for all x in an interval / then f(x) is decreasing on the interval 1. 3. If f' * (x) = 0 for all x in an interval / then f(x) is constant on the interval I. Concave Up/Concave Down 1. If f^ * (x) > 0 for all x in an interval I then f(x) is concave up on the interval 1. 2. If f' * (x) < 0 for all x in an interval / then f(x) is concave down on the interval I. Inflection Points x = c is a inflection point of f(x) if the concavity changes at x = c Extrema Absolute Extrema 1. x = c an absolute maximum of f(x) if f(c) >= f(x) for all x in the domain. 2. x = c is an absolute minimum of f(x) f(c) <= (f(x))! for all x in the domain. Fermat's Theorem If f(x) has a relative (or local) extrema at x = c then x = c a critical point of f(x) Extreme Value Theorem If f(x) is continuous on the closed interval [a, b] then there exist numbers c and d so that, 1. a <= c . 2. f(c) is the abs. max. in [a, b] 3. f(d) is the abs. min. in [a, b] Finding Absolute Extrema To find the absolute extrema of the continuous function f(x) on the interval [a, b] use the following process, 1. Find all critical points of f(x) in [a, b] 2. Evaluate f(x) at all points found in Step 1. 3. Evaluate f(a) and f(b) 4. Identify the abs. max. (largest function value) and the abs. min.(smallest function value) from the evaluations in Steps 2 & 3. Relative (local) Extrema 1. x = c is a relative (or local) maximum of f(x) if f(c) >= f(x) for all x near c. 2. x = c is a relative (or local) minimum of f(x) f(c) <= f(x) for all x near c. 1ª Derivative Test If x = c is a critical point of f(x) then x = c is 1. a rel. max. of f(x) if f' * (x) > 0 to the left of x = c and f' * (x) < 0 to the right of x = c 2. a rel. min. of f(x) if f' * (x) < 0 to the left x = c f' * (x) > 0 the right of x = c 3. not a relative extrema of f(x) if f' * (x) is the same sign on both sides of x = c 2 ^ (nd) Derivative Test If x = c is a critical point of f(x) such that f' * (c) = 0 then x = c 1. is a relative maximum of f(x) if f^ * (c) < 0 2. is a relative minimum of f(x) j' * (c) > 0 3. may be a relative maximum, relative minimum, or neither if f^ prime prime (c) = 0 Finding Relative Extrema and/or Classify Critical Points 1. Find all critical points of f(x) 2. Use the 1 ^ * pi derivative test or the 2 ^ (nd) derivative test on each critical point. Mean Value Theorem If f(x) is continuous on the closed interval [ [a, b] and differentiable on the open interval (a.b) then there is a number a < c < b such that f' * (c) = (f(b) - f(a))/(b - a) Newton's Method If x_{s} is the n ^ oplus guess for the root/solution of f(x) = 0 then (n - 1) ^ pi guess is provided f' * (x_{n}) exists. x n + 1 =x n - hat f (x n ) f^ prime (x n )