11. Partial Derivatives and Tangent Planes

d. Tangent Plane to the Graph of a Function

2. Algebraic Formula

Tangent Line to the Graph of a Function

Comparison of Tangent Line and Tangent Plane Derivations

a. Tangent Line to the Graph of a Function

In single variable calculus, the initial application of the derivative was to find an equation of the tangent line at a point on the graph of a function of \(1\) variable. Here is a review of that derivation.

We want to construct the tangent line at a point, \(x=a\), on the graph of a function, \(y=f(x)\). The most general line has the standard equation: \[ Ax+By=C. \] The line is vertical if \(B=0\) and non-vertical if \(B\ne0\). Assuming it is not vertical, we can solve for \(y\) and put the equation into slope-intercept form: \[ y=mx+b \qquad (1) \] where \(m\) is the slope and \(b\) is the \(y\)-intercept.

Now suppose we want to find the equation of the line tangent to \(y=f(x)\) at \(x=a\). We know the slope is \(m=f'(a)\). So equation \((1)\) becomes: \[ y=f'(a)x+b \qquad (2) \] We know the line passes through the point \((x,y)=(a,f(a))\). So equation \((2)\) tell us: \[ f(a)=f'(a)a+b \] or \[ b=f(a)-f'(a)a \] Using this formula for \(b\), equation \((2)\) becomes: \[\begin{aligned} y&=f'(a)x+f(a)-f'(a)a\\ &=f(a)+f'(a)(x-a) \end{aligned}\] which is the equation for the tangent line. We define the formula on the right to be the tangent function: \[ f_{\tan}(x)=f(a)+f'(a)(x-a) \] so that the equation of the tangent line is \(y=f_{\tan}(x)\).

The equation of the tangent line to the graph of the function \(y=f(x)\) at \(x=a\) is: \[ y=f_{\tan}(x) \equiv f(a)+f'(a)(x-a). \] In differential notation, this is: \[ y=f_{\tan}(x) \equiv f(a)+\left.\dfrac{df}{dx}\right|_{x=a}(x-a). \]

You have probably already memorized this.