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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=\ln (5 x+3 y), \quad A(2 ; 2), \quad \vec{a}=\{2 ;-3\} . \]

2.3.6 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=2 x^{3} y+3 x^{2} y^{2}, A(1 ;-2), \vec{a}=\{6 ;-8\} . \]

2.3.7 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=\sin ^{-1} \frac{x}{x+y}, \quad A(5 ; 5), \quad \vec{a}=\{-12 ; 5\} . \]

2.3.8 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=\frac{x+y}{x^{2}+y^{2}}, \quad A(1 ;-2), \quad \vec{a}=\{1 ;-2\} . \]

2.3.9 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given . It is required to find at point \( A \) : a) the gradient of the function and its value. b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=2 x^{4}+8 x^{2} y^{3}, A(2 ;-1), \quad \vec{a}=\{1 ;-3\} . \]

2.3.10 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given . It is required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=2 x^{2}+3 x y+4 y^{2}, A(2 ;-2), \quad \vec{a}=\{1 ; 3\} . \]

2.3.11 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given . It is required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=\tan ^{-1} \frac{y}{x}, \quad A(-1 ; 1), \quad \vec{a}=\{-1 ;-1\} . \]

2.3.12 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It's required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=\sqrt{x^{2}+y^{2}}, \quad A(3 ; 4), \quad \vec{a}=\{4 ;-3\} . \]

2.3.13 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=\tan ^{-1} \frac{y}{x}, \quad A(1 ; 3), \quad \vec{a}=\{3 ; 4\} . \]

2.3.14 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to find at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=x^{2}-x y+y^{2}, \quad A(2 ; 2), \quad \vec{a}=\{6 ; 8\} . \]

2.3.15 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to find a point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=x^{2}-y^{2}, \quad A(1 ; 1), \quad \vec{a}=\{2 ; 2\} . \]

2.3.16 Gradient and directional derivative

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Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to finc at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=\ln \left(x^{2}+y^{2}\right), \quad A(3 ; 4), \quad \vec{a}=\{2 ;-1\} . \]

2.3.17 Gradient and directional derivative

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Problem: Calculate the derivative of function \( f(x, y)=x^{3}+2 x^{2} y+2 \) at point \( A(2 ; 2) \quad \) in the direction of vector \( \overrightarrow{A B} \), where \( B=(4 ; 2) \).

2.3.18 Gradient and directional derivative

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Problem: Calculate the gradient of function \( z=x y \cos x y \) at point \( M(0 ;-3) \).

2.3.19 Gradient and directional derivative

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Problem: Find the gradient of function \[ f(x, y, z)=\frac{x^{2}+2 y}{2 y+z} \text { at point } A=(2 ; 4 ; 2) \text {. } \]

2.3.20 Gradient and directional derivative

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