11. Partial Derivatives and Tangent Planes

a. Computation of Partial Derivatives

Let's start with some simple calculations to introduce the concept of a partial derivative.

For a function of 2 variables \(f(x,y)\), we can take the derivative with respect to either variable while regarding the other variable as a constant. This is called a partial derivative. The prime notation for derivatives no longer makes sense because it does not say which is the variable of differentiation. So we replace it by a subscript notation: \(f'\) becomes \(f_x\) or \(f_y\). To further emphasize the difference between ordinary derivatives and partial derivatives we replace the “straight d,” \(d\), by a “curly d,” \(\partial\), which is read “partial.”

There are many ways to denote the derivative of a function of one variable, \(y=f(x)\), such as \[ f'(x)=\dfrac{df}{dx}=\dfrac{dy}{dx}=D(f)=D(y) \qquad \text{etc.} \] Likewise, there are multiple ways to refer to a partial derivative of a function of 2 variables, \(z=f(x,y)\). These include the following: \[f_x(x,y)=\dfrac{\partial f}{\partial x} =\dfrac{\partial z}{\partial x}=D_xf=D_xz =\partial_xf=\partial_xz \] \[f_y(x,y)=\dfrac{\partial f}{\partial y} =\dfrac{\partial z}{\partial y}=D_yf=D_yz =\partial_yf=\partial_yz \] In practice, the first three, \(f_x\), \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial z}{\partial x}\) are the most common. However, you should be comfortable using any of these notations.

In addition, there are many ways to denote the evaluation of a partial derivative at a point \((x,y)=(a,b)\) include the following: \[f_x(a,b)=\dfrac{\partial f}{\partial x}(a,b) =\left.\dfrac{\partial z}{\partial x}\right|_{(x,y)=(a,b)} =D_xf(a,b)=\left.\partial_xz\right|_{(a,b)} \]

Compute \(\dfrac{\partial f}{\partial y}\) for \(f=\ln(\sin(x^2+y^3))\). Then compute \(\left.\dfrac{\partial f}{\partial y}\right|_{(1,2)}\).

We regard \(x\) as a constant, because we are taking the partial derivative with respect to \(y\). We take the derivative just like we would any other derivative with constants. The function here is more complex than the previous examples, and we need to use the chain rule, twice. \[\begin{aligned} \dfrac{\partial f}{\partial y} &=\dfrac{1}{\sin(x^2+y^3)}\dfrac{\partial }{\partial y}\sin(x^2+y^3) =\dfrac{\cos(x^2+y^3)}{\sin(x^2+y^3)}\dfrac{\partial }{\partial y}(x^2+y^3) \\ &=\cot(x^2+y^3)(3y^2) =3y^2\cot(x^2+y^3) \end{aligned}\] In practice, you would not need to write all these steps, but rather go directly to \[ \dfrac{\partial f}{\partial y} =\dfrac{\cos(x^2+y^3)}{\sin(x^2+y^3)} 3 y^2 \] Finally, \[ \left.\dfrac{\partial f}{\partial y}\right|_{(1,2)} =3\cdot2^2 \cot(1^2+2^3)=12\cot9 \]