Functions of several variables: Gradient
Directional derivative
The first partial derivative \(f_x(a,b)\) of \(f(x,y)\) is the derivative of \(f(x,y)\) at \((a,b)\) in the direction of the positive \(x\) axis. Similarly, the first partial derivative\(f_y(a,b)\) in the derivative of \(f(x,y)\) at \((a,b)\) toward the positive \(y\) axis. We can also consider any direction in the tangent plane at \(\bigl(a,b,f(a,b)\bigr)\) on the surface graph of \(f\). If we take a tangent vector of length \(1\), then we get the following definition.
The directional derivative of \(f(x,y)\) at \((a,b)\) in the direction of \((a+{\vartriangle}x, b+{\vartriangle}y)\) is \[\frac{f_x(a,b){\vartriangle}x+f_y(a,b){\vartriangle}y}{\sqrt{{\vartriangle}x^2+{\vartriangle}y^2}}\]
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