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Hi mathematics,

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Finding y at x = 0

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Begin with:

x - y + sin(xy) = 0.

Let x = 0:

0 - y + sin(0y) = 0.

Simplify:

-y + sin(0) = 0.

Evaluate sin(0) = 0:

-y + 0 = 0.

Simplify:

-y = 0.

Multiply by -1:

y = 0.

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Finding dy/dx

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Begin with:

x - y + sin(xy) = 0.

Let xy = u:

x - y + sin(u) = 0.

Differentiate:

dx - dy + cos(u)du = 0.

Recall u = xy. Then differentiate by the product rule:

du = xdy + ydx.

Substitute the formulae for u and du:

dx - dy + cos(xy) (xdy + ydx) = 0.

Expand the brackets:

dx - dy + xcos(xy)dy + ycos(xy)dx = 0.

Move all dy to the RHS:

dx + ycos(xy)dx = dy - xcos(xy)dy.

Factorise the variable differentials on each side:

dx (1 + ycos(xy)) = dy (1 - xcos(xy)).

Cross divide:

dy/dx = (1 + ycos(xy)) / (1 - xcos(xy)).

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Calculating the gradient at (x,y) = (0,0)

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Substitute x = 0 and y = 0 into the formula for dy/dx:

dy/dx [0,0] = (1 + 0cos(0*0)) / (1 - 0cos(0*0)).

Simplify:

dy/dx [0,0] = (1 + 0) / (1 - 0).

Result:

dy/dx [0,0] = 1.

you need to find y first. plug x in the original, and y=0

0-y+sin(0) = 0

y=sin(0)

y=0

plug in both x and y into derivative and it will = 1

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