Partial Derivatives

A Partial derivative is defined as the derivative of the function w.r.t. one of the variables while holding the others constant ∂ f ∂ x = lim Δ x → 0 f ( x + Δ x , t ) − f ( x , t ) Δ x ∂ f ∂ x = lim Δ x → 0 f ( x + Δ x , t ) − f ( x , t ) Δ x ∂ f ∂ t = lim Δ t → 0 f ( x , t + Δ t ) − f ( x , t ) Δ t ∂ f ∂ t = lim Δ t → 0 f ( x , t + Δ t ) − f ( x , t ) Δ t Some examples: f ( x , t ) = 3 x 2 + x t 2 f ( x , t ) = 3 x 2 + x t 2 ∂ f ∂ x = 6 x + t 2 ∂ f ∂ x = 6 x + t 2 ∂ f ∂ t = 2 x t ∂ f ∂ t = 2 x t ∂ 2 f ∂ x 2 = 6 ∂ 2 f ∂ x 2 = 6 ∂ 2 f ∂ t 2 = 2 x ∂ 2 f ∂ t 2 = 2 x ∂ 2 f ∂ x ∂ t = ∂ 2 f ∂ t ∂ x = 2 t ∂ 2 f ∂ x ∂ t = ∂ 2 f ∂ t ∂ x = 2 t