What is the physical significance of partial derivative?
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Partial derivatives are just derivatives.
Consider a function f(x,y)f(x,y) of two variables. The partial derivative ∂f∂x∂f∂x is just the derivative when the other variable yy is taken to be a constant.
The graph z=f(x,y)z=f(x,y) is a surface in xyz-space. The tangent plane at a point (f(x0,y0),x0,y0)(f(x0,y0),x0,y0) to the surface has two slopes: the slope in the x-direction, which is ∂f∂x∂f∂x evaluated at (x0,y0)(x0,y0) and the slope in the y-direction, which is ∂f∂y∂f∂y evaluated at (x0,y0)(x0,y0).
If you intersect the graph z=f(x,y)z=f(x,y) with the vertical plane y=y0y=y0, the result is a curve in that plane. The slope of that curve is the slope in the x-direction mentioned in the previous line.
Consider a function f(x,y)f(x,y) of two variables. The partial derivative ∂f∂x∂f∂x is just the derivative when the other variable yy is taken to be a constant.
The graph z=f(x,y)z=f(x,y) is a surface in xyz-space. The tangent plane at a point (f(x0,y0),x0,y0)(f(x0,y0),x0,y0) to the surface has two slopes: the slope in the x-direction, which is ∂f∂x∂f∂x evaluated at (x0,y0)(x0,y0) and the slope in the y-direction, which is ∂f∂y∂f∂y evaluated at (x0,y0)(x0,y0).
If you intersect the graph z=f(x,y)z=f(x,y) with the vertical plane y=y0y=y0, the result is a curve in that plane. The slope of that curve is the slope in the x-direction mentioned in the previous line.
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