Taylor's Theorem Analogous to the one-dimensional Taylor’s theorem .
Suppose f : R 2 → R f: \mathbb{R}^2 \to \mathbb{R} f : R 2 → R If all partial derivatives of order n n n n + 1 n+1 n + 1 then :
f ( x 0 + h , y 0 + k ) = f ( x 0 , y 0 ) + ∑ i = 1 n 1 i ! ( h ∂ ∂ x + k ∂ ∂ y ) i f ∣ ( x 0 , y 0 ) + R n f(x_0 + h, y_0 + k) = f(x_0, y_0) +
\sum_{i=1}^{n} \frac{1}{i!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^i f \Bigg|_{(x_0, y_0)} +
R_n f ( x 0 + h , y 0 + k ) = f ( x 0 , y 0 ) + i = 1 ∑ n i ! 1 ( h ∂ x ∂ + k ∂ y ∂ ) i f ( x 0 , y 0 ) + R n Here
R n = 1 ( n + 1 ) ! ( h ∂ ∂ x + k ∂ ∂ y ) n + 1 f ∣ ( x 0 + h , y 0 + k ) R_n = \frac{1}{(n+1)!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^{n+1} f \Bigg|_{(x_0+h, y_0+k)} R n = ( n + 1 )! 1 ( h ∂ x ∂ + k ∂ y ∂ ) n + 1 f ( x 0 + h , y 0 + k ) And θ ∈ ( 0 , 1 ) \theta \in (0,1) θ ∈ ( 0 , 1 )
( h ∂ ∂ x + k ∂ ∂ y ) n \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^n ( h ∂ x ∂ + k ∂ y ∂ ) n Taylor Series
f ( x , y ) = ∑ n = 0 ∞ { 1 n ! ∑ k = 0 n ( n k ) ∂ n f ∂ x n − k ∂ y k ∣ ( x 0 , y 0 ) ( x − x 0 ) n − k ( y − y 0 ) k } f(x, y) = \sum_{n=0}^{\infty} \left\{ \frac{1}{n!} \sum_{k=0}^{n} \binom{n}{k} \frac{\partial^n f}{\partial x^{n-k} \partial y^k} \bigg|_{(x_0, y_0)} (x - x_0)^{n-k} (y - y_0)^k \right\} f ( x , y ) = n = 0 ∑ ∞ { n ! 1 k = 0 ∑ n ( k n ) ∂ x n − k ∂ y k ∂ n f ( x 0 , y 0 ) ( x − x 0 ) n − k ( y − y 0 ) k } Linear & Quadratic Approximations
Linear approximation for a function f f f
f ( x 0 + h , y 0 + k ) ≈ f ( x 0 , y 0 ) + h ∂ f ∂ x ∣ ( x 0 , y 0 ) + k ∂ f ∂ y ∣ ( x 0 , y 0 ) f(x_0 + h, y_0 + k) \approx f(x_0, y_0) + h\frac{\partial f}{\partial x} \bigg|_{(x_0, y_0)} + k\frac{\partial f}{\partial y} \bigg|_{(x_0, y_0)} f ( x 0 + h , y 0 + k ) ≈ f ( x 0 , y 0 ) + h ∂ x ∂ f ( x 0 , y 0 ) + k ∂ y ∂ f ( x 0 , y 0 ) Quadratic approximation for a function f f f
f ( x 0 + h , y 0 + k ) ≈ f ( x 0 , y 0 ) + h ∂ f ∂ x ∣ ( x 0 , y 0 ) + k ∂ f ∂ y ∣ ( x 0 , y 0 ) + 1 2 ( h 2 ∂ 2 f ∂ x 2 ∣ ( x 0 , y 0 ) + 2 h k ∂ 2 f ∂ x ∂ y ∣ ( x 0 , y 0 ) + k 2 ∂ 2 f ∂ y 2 ∣ ( x 0 , y 0 ) ) f(x_0 + h, y_0 + k)
\approx f(x_0, y_0) +
h\frac{\partial f}{\partial x} \bigg|_{(x_0, y_0)} +
k \frac{\partial f}{\partial y} \bigg|_{(x_0, y_0)} +
\frac{1}{2} \left(h^2 \frac{\partial^2 f}{\partial x^2} \bigg|_{(x_0, y_0)} + 2hk \frac{\partial^2 f}{\partial x \partial y} \bigg|_{(x_0, y_0)} + k^2 \frac{\partial^2 f}{\partial y^2} \bigg|_{(x_0, y_0)} \right) f ( x 0 + h , y 0 + k ) ≈ f ( x 0 , y 0 ) + h ∂ x ∂ f ( x 0 , y 0 ) + k ∂ y ∂ f ( x 0 , y 0 ) + 2 1 ( h 2 ∂ x 2 ∂ 2 f ( x 0 , y 0 ) + 2 hk ∂ x ∂ y ∂ 2 f ( x 0 , y 0 ) + k 2 ∂ y 2 ∂ 2 f ( x 0 , y 0 ) ) Written by September 29, 2026 · 1 min read ·
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