11. Partial Derivatives and Tangent Planes

d. Tangent Plane to the Graph of a Function

2. Algebraic Formula

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 $y=f_{\tan}(x)$.

Equation of a Tangent Line to a Graph
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.