Answer to Question #277966 in Calculus for Nick

Question

2. a) Find the derivatives of the following functions with respect to x.

x ^ 3 + y ^ 3 = 3

y = (sin x) ^ tan x

b) Evaluate the 2 ^ (nd) order partial derivatives partial^ 2 u partial x^ 2 and partial^ 2 u partial y^ 2 if u=2x^ 3 +3x^ 2 y+xy^ +y^ .

Answer 1

$\displaystyle x^3 +y^3 =3\\ y^3 = 3-x^3\\ \text{Differentiating implicitly, we get}\\ 3y^2\frac{dy}{dx}=-3x^2\\ \frac{dy}{dx}= -\frac{x^2}{y^2}\\ y =\sin(x)^{\tan(x)}\\ \text{We take the natural logarithm to obtain}\\ \ln y = \tan x \ln \sin x\\ \text{Next, we take the exponential of both sides to obtain}\\ \frac{1}{y}\cdot\frac{dy}{dx} = \sec^2x\ln \sin x + \tan x \frac{1}{\sin x}\cdot \cos x\\ = 1 + \ln \sin x \sec^2x\\ \implies \frac{1}{y}\cdot\frac{dy}{dx} = = 1 + \ln \sin x \sec^2x\\ \text{Multiplying both sides by $y = \sin x^{\tan x}$, we have}\\ \frac{d(\sin x^{\tan x})}{dx}= (1+ \ln \sin x \sec^2x)\sin x^{\tan x}\\ u = 2x^3 + 3x^2y+xy^2+y^2\\ \text{The 2nd order partial derivatives is given by $u_{xx}, u_{xy}, u_{yy}$}\\ u_{xx}= 12x + 6y\\ u_{xy}= 6x+2y\\ u_{yx} = 6x+2y\\ u_{yy} = 2x + 2$