Is the gradient of a curve at a given point the same as the gradient of the tangent of the curve at that given point?

Yes the gradient of a point that is on a given curve is the same as the tangent of the curve at that point. Hence you find the both gradient of the curve and the tangent by differentiating the function of the curve and substituting the values of that point into the resulting differential to obtain a value of its gradient function dy/dx. (Note that if you require implicit differentiation for this process you will need both the value of x and y to substitute at that point on the differential you have found and then rearrange to find dy/dx).