What is the gradient of y = xcos(x) at x=0?
First we want to calculate the derivative of y. To do this we use the product rule:If we rewrite y as y = uv, then dy/dx = vdu/dx + udv/dx.Here, we have u = x and v = cos(x).That means du/dx = 1 and dv/dx = -sin(x).Therfore dy/dx = cos(x)1 + x(-sin(x)) = cos(x) - xsin(x).To evaluate the gradient of y at x=0 we substitute x=0 into the derivative we have just calculated:gradient = cos(0) - 0*sin(0) = 1
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