14. Directional Derivatives and Gradients

b. Derivative along a Vector

When we introduced the limit definition of partial derivatives, we also gave the definition of the derivative of a function \(f(x,y)\) along an arbitrary vector \(\vec v\):

Given any vector \(\vec v\), the derivative of \(f(x,y)\) along \(\vec v\) (or with respect to \(\vec v\)) is: \[ \nabla_{\vec v}f=\lim_{h\rightarrow 0} \dfrac{f((x,y)+h\vec v)-f(x,y)}{h} \]

Now that we have covered the chain rule, we can give a formula for the derivative along a vector that does not involve limits.

Given any vector \(\vec v\), the derivative of \(f(x,y)\) along \(\vec v\) (or with respect to \(\vec v\)) is: \[ \nabla_{\vec v}f=\vec v\cdot\vec\nabla f \] The proof is easy. Try reading it!

Given a vector \(\vec v\), consider the parametric curve \(\vec r(t)=(x,y)+t\vec v\). We will compute the derivative of the composition \[ f(t)=(f\circ\vec r)(t)=f(\vec r(t)) \] at \(t=0\) in two ways. On the one hand, the derivative of the composition is the derivative along the curve as discussed on the previous page: \[ \left.\dfrac{df}{dt}\right|_{t=0} =\left.\dfrac{d(f\circ\vec r)}{dt}\right|_{t=0} =\vec v\cdot\left.\vec\nabla f\right|_{\vec r(0)} =\vec v\cdot\left.\vec\nabla f\right|_{(x,y)} \] On the other hand, since the composition is a function of \(1\) variable, we can compute its derivative using the limit definition: \[\begin{aligned} \left.\dfrac{df}{dt}\right|_{t=0} =\left.\dfrac{d(f\circ\vec r)}{dt}\right|_{t=0} &=\lim_{h\rightarrow 0} \dfrac{(f\circ\vec r)(0+h)-(f\circ\vec r)(0)}{h}\\ &=\lim_{h\rightarrow 0} \dfrac{(f(\vec r(h))-(f(\vec r(0))}{h}\\ &=\lim_{h\rightarrow 0} \dfrac{f((x,y)+h\vec v)-f(x,y)}{h} \end{aligned}\] But this is the limit of \(\nabla_{\vec v}f\). Consequently: \[ \nabla_{\vec v}f=\vec v\cdot\vec\nabla f \]

As a simple consequence, we have:

The derivative of \(f(x,y)\) along a curve \(\vec r(t)\) is: \[ \dfrac{df}{dt}=\dfrac{d(f\circ\vec r)}{dt} =\vec v\cdot\vec\nabla f=\nabla_{\vec v}f \] In words, the derivative of \(f(x,y)\) along a curve \(\vec r(t)\) is the derivative of \(f(x,y)\) along its velocity (tangent vector) \(\vec v\). Conversely, the derivative of \(f(x,y)\) along a vector \(\vec v\) is the derivative of \(f(x,y)\) along any curve which passes through \((x,y)\) with velocity \(\vec v\).

Find the derivative of \(f(x,y)=\dfrac{x}{y}\) along \(\vec v=\langle 3,4\rangle\) at the point \(P=(1,2)\).

The gradient of \(f\) is: \[ \vec\nabla f =\left\langle \dfrac{\partial f}{\partial x}, \dfrac{\partial f}{\partial y}\right\rangle =\left\langle \dfrac{1}{y},-\,\dfrac{x}{y^2}\right\rangle \] At the point \(P\) this is: \[ \left.\vec\nabla f\right|_P =\left\langle \dfrac{1}{2},-\,\dfrac{1}{4}\right\rangle \] So the derivative of \(f\) along \(\vec v\) is: \[ \left.\vec\nabla_{\vec v}f\right|_P =\vec v\cdot\left.\vec\nabla_f\right|_P =\langle 3,4\rangle\cdot\left\langle \dfrac{1}{2},-\,\dfrac{1}{4}\right\rangle =\dfrac{3}{2}-1 =\dfrac{1}{2} \]

Find the rate of change of \(f=\dfrac{x}{y}\) along the curve \(\vec r(t)=(1+3t,2+4t)\) at \(t=0\). (Note that this is a straight line.)

Since the point is \(P=\vec r(0)=(1,2)\) and the velocity is \(\vec v=\dfrac{d\vec r}{dt}=\langle 3,4\rangle\), the rate of change is the same as the derivative in the previous example. \[ \dfrac{df}{dt}=\left.\vec\nabla_{\vec v}f\right|_P =\dfrac{1}{2} \]

Find the rate of change of \(f=\dfrac{x}{y}\) along the curve \(\vec r(t)=(t^3,2 t^2)\) at \(t=1\). (Note that this is not a straight line.)

Since the point is \(P=\vec r(1)=(1,2)\) and the velocity is \(\vec v=\left.\dfrac{d\vec r}{dt}\right|_{t=1} =\left.\langle 3t^2,4t\rangle\right|_{t=1} =\langle 3,4\rangle\), the rate of change is the same as in the previous example. \[ \dfrac{df}{dt}=\left.\vec\nabla_{\vec v}f\right|_P =\dfrac{1}{2} \]

Find the derivative of \(P=z(2-\cos x)(2+\sin y)\) along \(\vec v=\langle 3,2,1\rangle\) at the point \(X=(x,y,z)=(\dfrac{\pi}{2},\pi,2)\).

\(\nabla_{\vec v}P=8\)

The gradient of \(P\) is \[\begin{aligned} \vec\nabla P &=\left\langle z(\sin x)(2+\sin y), z(2-\cos x)(\cos y),\right. \\ &\qquad\left.(2-\cos x)(2+\sin y)\right\rangle \end{aligned}\] At the point \(X\) this is \[\begin{aligned} \left.\vec\nabla P\right|_{(\pi/2,\pi,2)} &=\left\langle 2\left(\sin\dfrac{\pi}{2}\right)(2+\sin\pi), 2\left(2-\cos\dfrac{\pi}{2}\right)(\cos\pi),\right. \\ &\qquad\left.\left(2-\cos\dfrac{\pi}{2}\right)(2+\sin\pi)\right\rangle \\ &=\langle 4,-4,4\rangle \end{aligned}\] So, the derivative of \(P\) along \(\vec v\) is: \[ \nabla_{\vec v}P =\vec v\cdot\left.\vec\nabla P\right|_{(\pi/2,\pi,2)} =\langle 3,2,1\rangle\cdot\langle 4,-4,4\rangle =8 \]

Notice that this exercise is essentially the same as the last exercise on the previous page except the first is along a curve and the second is along a vector which happens to be the velocity of that curve.

14. Directional Derivatives and Gradients

b. Derivative along a Vector

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