Find all the first and second order partial derivatives of f(x,y)=(x^2+2y)\cos(xy).

Question:

Find all the first and second order partial derivatives of {eq}f(x,y)=(x^2+2y)\cos(xy){/eq}.

Partial Derivative

The derivative of a function containing two or more variable with respect to one taking other as the constant is known as the partial derivative.

Answer and Explanation:

{eq}\Rightarrow \ f(x,y)=(x^{2}+2y)\cos(xy)\\ \text{diffrentiate with respect to x taking y as a contant to the partial derivative with respect to x }\\ \Rightarrow \ f'(x,y)=\frac{d}{dx}((x^{2}+2y)\cos(xy))\\ \Rightarrow \ f'(x,y)=((x^{2}+2y)\frac{d}{dx}cos(xy)+cos(xy)\frac{dx}{dx}(x^{2}+2y)\\ \Rightarrow \ f'(x,y)=-(x^{2}+2y)sin(xy)+2xcos(xy)\\ \text{differentiate } f'(x,y) \text{ with respect to x taking y as costant to get } f"(x,y) \\ \Rightarrow \ f"(x,y)=-sin(xy)\frac{d}{dx}(x^{2}+2y)-(x^{2}+2y)\frac{d}{dx}sin(xy)+\frac{d}{dx}2xcos(xy)\\ \Rightarrow \ f"(x,y)=-2xsin(xy)-cos(xy)(x^{2}+2y)-2sin(xy)\\ \Rightarrow \ f(x,y)=(x^{2}+2y)\cos(xy)\\ \text{Differentiate with respect to y taking x as contant}\\ \Rightarrow \ f'(x,y)=\frac{d}{dy}((x^{2}+2y)\cos(xy))\\ \Rightarrow \ f'(x,y)=((x^{2}+2y)\frac{d}{dy}cos(xy)+cos(xy)\frac{dx}{dy}(x^{2}+2y)\\ \Rightarrow \ f'(x,y)=-(x^{2}+2y)sin(xy)+2cos(xy)\\ \text{differentiate } f'(x,y) \text{ with respect to y taking x as costant to get } f"(x,y) \\ \Rightarrow \ f"(x,y)=-sin(xy)\frac{d}{dy}(x^{2}+2y)-(x^{2}+2y)\frac{d}{dy}sin(xy)+\frac{d}{dy}2cos(xy)\\ \Rightarrow \ f"(x,y)=-2sin(xy)-sin(xy)(x^{2}+2y)-2sin(xy)\\ \Rightarrow \ f"(x,y)=-4sin(xy)-sin(xy)(x^{2}+2y)\\ {/eq}


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Evaluating Definite Integrals Using the Fundamental Theorem

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