Derivatives and differentials

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Problem: Calculate the second order differential of the function \( z=x y \cos x y \).

2.2.16 Derivatives and differentials

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Problem: Calculate the partial derivatives of the function of three variables, given parametrically: \[ \begin{array}{l} u=\ln (2 x-\sqrt{z})+\frac{1}{5} \tan ^{-1} \frac{y}{x} \\ x=x(s), y=y(t), z=z(t, s) . \end{array} \]

2.2.17 Derivatives and differentials

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Problem: Find derivatives \( \frac{d y}{d x} \) of the following functions: 1. \( y=\operatorname{ch}(\sin x) \). 2. Implicitly defined: \( 5 x^{4}+3 y-7=0 \).

2.2.18 Derivatives and differentials

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Problem: Find derivatives \( \frac{d y}{d x} \) and \( \frac{d^{2} y}{d x^{2}} \) for the given functions: 1. \( y=\ln \cot 2 x \). 2. \( x=t^{3}+8 t, y=t^{5}+2 t \).

2.2.19 Derivatives and differentials

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Problem: Find \( \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y} \), if functions \( u \) and \( v \) are implicitly defined by the equalities \[ x=e^{u}+u \sin v, y=e^{u}-u \cos v . \]

2.2.20 Derivatives and differentials

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Problem: Let \( u=\frac{1}{x}(\varphi(x-y)+\psi(x+y)) \), where \( \varphi \) and \( \psi \) are differentiable functions. Show that \[ \frac{\partial}{\partial x}\left(x^{2} \frac{\partial u}{\partial x}\right)=x^{2} \frac{\partial^{2} u}{\partial y^{2}} . \]

2.2.21 Derivatives and differentials

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Problem: Calculate the derivative of the function, given parametrically: \[ \left\{\begin{array}{l} x=\cot t \\ y=\cos t \end{array}\right. \]

2.2.22 Derivatives and differentials

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Problem: Find the derivatives of the functions: a) \( y=3 \sqrt[3]{\frac{x+1}{(x-1)^{2}}} \), b) \( y=x-3 \ln \left[\left(1+e^{\frac{x}{6}}\right) \sqrt{1+e^{\frac{x}{3}}}\right. \), c) \( y=\frac{1}{2} \sqrt{\frac{1}{x^{2}}-1}-\frac{\arccos x}{2 x^{2}} \), d) \( y=\ln \left(\arccos \frac{1}{\sqrt{x}}\right) \), e) \( y=\frac{6^{x}(\sin 4 x \ln 6-4 \cos 4 x)}{16+\ln ^{2} 6} \), f) \( x=\left(1+\cos ^{2} t\right)^{2}, y=\frac{\cos t}{\sin ^{2} t} \).

2.2.23 Derivatives and differentials

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Problem: Find the derivatives of the functions: a) \( y=\frac{1+x^{2}}{2 \sqrt{1+2 x^{2}}} \), b) \( y=e^{a x}\left[\frac{1}{2 a}+\frac{a \cos 2 b x+2 b \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right] \), c) \( y=\ln \sin \frac{2 x+4}{x+1} \) d) \( y=\frac{4+x^{4}}{x^{3}} \arctan \frac{x^{2}}{2}+\frac{4}{x} \), e) \( y=\left(x^{3}+4\right)^{\tan x} \), f) \( x=\arcsin \sqrt{1-t^{2}}, \quad y=(\arccos t)^{2} \).

2.2.24 Derivatives and differentials

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