Derivatives and differentials

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Problem: Check that function \( U=\ln \frac{1}{\sqrt{x^{2}+y^{2}}} \) is a solution to the partial differential equation \( \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 \).

2.2.1 Derivatives and differentials

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Problem: Calculate the derivative of the complex function: \[ f(x)=\tan ^{-1}\left(\frac{\ln (x+2)}{\sin \sqrt{2 x}}\right), g(x)=\sqrt{\cos 4 x} . \]

2.2.2 Derivatives and differentials

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Problem: Calculate the derivative of the implicitly defined function \( y^{2}-\sin (2 x+y)=x^{3} \).

2.2.3 Derivatives and differentials

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Problem: Find the first and second order partial derivatives of the function: \[ z=\frac{1}{\tan ^{-1} \frac{y}{x}} . \]

2.2.4 Derivatives and differentials

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Problem: Find the first and second order partial derivatives of the function: \[ z=\ln \left(\frac{1}{\sqrt[3]{x}}-\frac{1}{\sqrt[3]{y}}\right) \]

2.2.5 Derivatives and differentials

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Problem: Find the first and second order total differentials of the function: \[ \omega=\sin ^{-1} \frac{x}{y} \text {. } \]

2.2.6 Derivatives and differentials

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Problem: Approximately calculate (using the differential): \[ A=(1,02)^{2} \cdot(0,97)^{2} \text {. } \]

2.2.7 Derivatives and differentials

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Problem: Find the partial derivatives and the differentials of the first and second order: \[ z=\tan ^{-1} \frac{y}{x} . \]

2.2.8 Derivatives and differentials

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Problem: A complex function is given: \[ z=\frac{y}{x} \text {, where } x=e^{t}, y=1-e^{2 t} \text {. } \] Find \( \frac{d z}{d t}, \quad \frac{d^{2} z}{d t^{2}} \).

2.2.9 Derivatives and differentials

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Problem: Find the first and second order derivatives of the implicit function: \( x^{2}+y^{2}-4 x+6 y=0 \).

2.2.10 Derivatives and differentials

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Problem: Function \( z=f(x, y) \) is given. Show that \[ \begin{array}{l} F\left(x ; y ; z ; \frac{\partial z}{\partial x} ; \frac{\partial z}{\partial y} ; \frac{\partial^{2} z}{\partial x^{2}} ; \frac{\partial^{2} z}{\partial y^{2}} ; \frac{\partial^{2} z}{\partial x \partial y}\right)=0 . \\ z=\frac{y^{2}}{3 x}+\sin ^{-1}(x y), \quad F=x^{2} \frac{\partial z}{\partial x}-x y \frac{\partial z}{\partial y}+y^{2} . \end{array} \]

2.2.11 Derivatives and differentials

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Problem: Check the equality of the mixed derivatives of the function \( z=e^{x+y} \cdot(x \cdot \cos y+y \cdot \sin x) \).

2.2.12 Derivatives and differentials

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Problem: Calculate the derivative of the function of two variables, given parametrically: \[ z=z(x, y), x=e^{1+y}, y=(1+e)^{t} . \]

2.2.13 Derivatives and differentials

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Problem: Calculate the partial derivatives of the function of two variables, given parametrically: \[ z=z(x, y), x=u^{3}+v^{3}, y=u^{2}+\ln v . \]

2.2.14 Derivatives and differentials

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Problem: Calculate the differential of the function \[ z=x y \cos x y . \]

2.2.15 Derivatives and differentials

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