Math I: Derivatives PDF

# Math I: Derivatives ## Hyperbolic Functions 1. _y = Sinh x_ : _y' = Cosh x_ 2. _y = Cosh x_ : _y' = Sinh x_ 3. _y = tanh x_ : _y'= Sech x_ 4. _y = Sech x_ : _y' = -Sech x tanh x_ 5. _y = Csch x_ : _y' = -Csch x Coth x_ 6. _y = Coth x_ : _y' = -Csch^2 x_ ## Generalization of the rules _y...

# Math I: Derivatives ## Hyperbolic Functions 1. _y = Sinh x_ : _y' = Cosh x_ 2. _y = Cosh x_ : _y' = Sinh x_ 3. _y = tanh x_ : _y'= Sech x_ 4. _y = Sech x_ : _y' = -Sech x tanh x_ 5. _y = Csch x_ : _y' = -Csch x Coth x_ 6. _y = Coth x_ : _y' = -Csch^2 x_ ## Generalization of the rules _y = Sinh f(x)_ _y' = f' (x) Cosh f(x)_ ## Examples ### Example 1 _y = Sinh (x + 7)_ _y' = 2x Cosh (x + 7)_ ### Example 2 _y = tanh (x^2 + 1)_ _y' = e^x Sech^2 (x^2 + 1)_ ### Example 3 _y = Sinh^3 x_ _y' = 3 (Sinh^2 x) * Cosh x_ _y' = 3 Sinh^2 x Cos h x_ ### Example 4 _y = √(Sin x + Sinh x)_ _y'= (Cos x + Cosh x) / (2√(Sin x + Sinh x))_ ### Example 5 _y = e^x Sinh x_ _y' = e^x (Sinh x + Cosh x)_ ### Example 6 _y = ln (1 + tanh x)_ _y' = Sech^2 x / (1 + tanh x)_ ### Example 7 _y = Cosh x / e_ _y' = Cosh x / e^x_ ### Example 8 _y = (1 + Sinh x) / (1 + Cosh x)_ _y' = (Cosh x (1 + Cosh x) - Sinh x (1 + Sinh x)) / (1 + Cosh x)^2_ _y' = (Cosh x + Cosh^2 x - Sinh x - Sinh^2 x) / (1 + Cosh x)^2_ _y' = (1 + Cosh x - Sinh x) / (1 + Cosh x)^2_ ### Example 9 _y = (1 + tanh x)^4_ _y' = 4 (1 + tanh x)^ 3 * Sech^2 x_ ### Example 10 _y = ln (tan x)_ _y' = Sec^2 x / tan x_ ### Example 11 _y = ln (cot x)_ _y' = -Csc^2 x / cot x_ ### Example 12 _y = ln (tan h x)_ _y' = Sech^ 2x / tanh x_ ## Inverse Functions ### Example 1 _y = Sin^-1 x_ _y' = 1/(√(1 - x^2))_ ### Example 2 _y = Cos^-1 (e^x)_ _y' = -1/(√(1-e^(2x))) * e^x_ ### Example 3 _y = tan^-1 (ln x)_ _y' = 1/(1 + (ln x)^2) * 1/x_ ### Example 4 _y = Cot^-1 (ln x)_ _y' = -1/(1 + (ln x)^2) * 1/x_ _y' = -1/(x(1 + (ln x)^2))_ ### Example 5 _y = Sec^-1 (Sin x)_ _y' = (Cos x) / ((Sin x)√(Sin^2 x - 1))_ ### Example 6 _y = Sec^-1 (ln x)_ _y' = 1/((ln x)√(ln^2 x - 1)) * 1/x_ _y' = 1/((x ln x)√(ln^2 x - 1))_ ### Example 7 _y = Sin^-1 (tan x)_ _y' = Sec x / (√(1 - tan^2 x))_ ### Example 8 _y = Cot^-1 (tan x)_ _y' = Sec^ 2x/ (1 + tan^2 x)_ ### Example 9 _y = Cos^-1 (Sin x)_ _y' = -Cos x / (√(1 - Sin^2 x))_ ## Derivatives ### Normal _y = P(x)_ _y' = P'(x)_ ### Implicit _g(x, y) = 0_ _y * g'(x, y) = -x * g'(x, y)_ ### Parametric _x = y(t)_ _y = g(t)_ _y' = g'(t) / y'(t)_ ### Logarithmic _y = f(x)_ _y' = (f'(x)) / (f(x))_ ## Implicit Differentiation _y = y(x)_ _y' = y'(x)_ _g(x, y) = 0_ _x'_**_ + y'_**_ * y'(x, y) = 0_ ## Examples ### Example 1 _x^2 + x * y + y^2 = 7_ _2x + y + x * y' + 2y * y' = 0_ _y' (x + 2y) = -2x - y_ _ y' = (-2x - y) / (x + 2y)_