Effect of Increment of Variable on Output in a Multivariate Equation

There is an equation as you can see below. Here, C1, C2 and C3 denote constants, while x and y denote variables.

I want to observe the effect of x on U. In short, will increasing x increase or decrease U? How can I prove this mathematically? I tried using limit but it didn't work because of two different variables. In addition, a constraint such as 40<x<55 can be set for x.

[Dyuman Joshi]

I don't see how this is related to MATLAB.

Answer 1

[Star Strider]

Perhaps taking the partial derivative of U with respect to x (creating a sort of sensitivity function) will do what you want. You can do this symbolically using the diff function:, or numerically using the gradient function.

EDIT — .(11 Aug 2023 at 15:39)

In detail —

syms C_1 C_2 C_3 x y 
sympref('AbbreviateOutput',false);
U = (C_1 * y * tan(x+(y/2))) / (C_2 * tan(x+(y/2)) + C_3 * (cos(y)+tan(x)*sin(y) - 1))

U = 

dUdx = diff(U,x);
dUdx = simplify(dUdx, 500)

dUdx = 

dUdxfcn = matlabFunction(dUdx)

dUdxfcn = function_handle with value:
    @(C_1,C_2,C_3,x,y)(C_1.*y.*(tan(x+y./2.0).^2+1.0))./(C_3.*(cos(y)+tan(x).*sin(y)-1.0)+C_2.*tan(x+y./2.0))-C_1.*y.*tan(x+y./2.0).*(C_2.*(tan(x+y./2.0).^2+1.0)+C_3.*sin(y).*(tan(x).^2+1.0)).*1.0./(C_3.*(cos(y)+tan(x).*sin(y)-1.0)+C_2.*tan(x+y./2.0)).^2

Supply the appropriate variable values and an appropriate vector for ‘x’ to evaluate the function, then plot the evaluated ‘dUdxfcn’ as a function of ‘x’.

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