Differentials and Partial Derivatives — Formula Sheet

Differentials and Partial Derivatives — Formula Sheet

One-variable approximation and differentials

• Linear approximation: \( L(x) = f(x_{0}) + f'(x_{0})(x - x_{0}) \)
• Approximation rule: \( f(x_{0} + \Delta x) \approx f(x_{0}) + f'(x_{0})\,\Delta x \)
• Differential: \( dy = f'(x)\,dx \)
• Actual change: \( \Delta y = f(x + \Delta x) - f(x) \), with \( \Delta y \approx dy \) for small \( \Delta x \)

Errors

• Absolute error \( = \lvert \text{actual} - \text{approximate} \rvert \)
• Relative error \( = \dfrac{\lvert \text{actual} - \text{approximate} \rvert}{\lvert \text{actual} \rvert} \)
• Percentage error \( = \text{relative error} \times 100 \)
• Power-law shortcut: if \( Q = k\,x^{n} \) then \( \dfrac{dQ}{Q} = n\,\dfrac{dx}{x} \)

Selected differentials

• \( d(x^{n}) = n x^{n-1}\,dx \)
• \( d(\sin x) = \cos x\,dx \), \( \quad d(\cos x) = -\sin x\,dx \)
• \( d(e^{x}) = e^{x}\,dx \), \( \quad d(\log x) = \dfrac{1}{x}\,dx \)
• \( d(\tan x) = \sec^{2} x\,dx \)

Properties of differentials

• \( d(c) = 0 \) (constant), \( \quad d(x) = dx \)
• \( d(cf) = c\,df \)
• \( d(f \pm g) = df \pm dg \)
• Product: \( d(fg) = f\,dg + g\,df \)
• Quotient: \( d\!\left(\dfrac{f}{g}\right) = \dfrac{g\,df - f\,dg}{g^{2}} \), \( \ g \neq 0 \)

Partial derivatives (definitions)

• \( \dfrac{\partial F}{\partial x}(x_{0},y_{0}) = \displaystyle\lim_{h\to 0}\dfrac{F(x_{0}+h,\,y_{0}) - F(x_{0},y_{0})}{h} \)
• \( \dfrac{\partial F}{\partial y}(x_{0},y_{0}) = \displaystyle\lim_{k\to 0}\dfrac{F(x_{0},\,y_{0}+k) - F(x_{0},y_{0})}{k} \)
• Second order: \( F_{xx} = \dfrac{\partial^{2}F}{\partial x^{2}},\ F_{yy} = \dfrac{\partial^{2}F}{\partial y^{2}},\ F_{xy} = \dfrac{\partial^{2}F}{\partial y\,\partial x} \)
• Clairaut’s theorem: if \( F_{xy}, F_{yx} \) are continuous, then \( F_{xy} = F_{yx} \)
• Laplace’s equation (harmonic): \( \dfrac{\partial^{2}u}{\partial x^{2}} + \dfrac{\partial^{2}u}{\partial y^{2}} = 0 \)

Several-variable approximation and differential

• Tangent plane: \( L(x,y) = F(x_{0},y_{0}) + F_{x}(x_{0},y_{0})(x - x_{0}) + F_{y}(x_{0},y_{0})(y - y_{0}) \)
• Total differential (2 vars): \( dF = \dfrac{\partial F}{\partial x}\,dx + \dfrac{\partial F}{\partial y}\,dy \)
• Total differential (3 vars): \( dF = \dfrac{\partial F}{\partial x}\,dx + \dfrac{\partial F}{\partial y}\,dy + \dfrac{\partial F}{\partial z}\,dz \)

Chain rule

• One parameter: \( \dfrac{dw}{dt} = \dfrac{\partial w}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial w}{\partial y}\dfrac{dy}{dt} \)
• Two parameters: \( \dfrac{\partial w}{\partial s} = \dfrac{\partial w}{\partial x}\dfrac{\partial x}{\partial s} + \dfrac{\partial w}{\partial y}\dfrac{\partial y}{\partial s} \), and similarly for \( \dfrac{\partial w}{\partial t} \)

Homogeneous functions and Euler’s theorem

• Homogeneous of degree \(p\): \( F(tx, ty) = t^{p} F(x,y) \)
• Euler (2 vars): \( x\,\dfrac{\partial F}{\partial x} + y\,\dfrac{\partial F}{\partial y} = p\,F \)
• Euler (3 vars): \( x\,\dfrac{\partial F}{\partial x} + y\,\dfrac{\partial F}{\partial y} + z\,\dfrac{\partial F}{\partial z} = p\,F \)