Tangent Lines & The Derivative as a Limit Cheat Sheet

A printable reference covering tangent lines, secant slopes, difference quotients, derivative limits, and point-slope form for grades 11-12.

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This cheat sheet covers how tangent lines are connected to derivatives and limits. Students need it because many calculus problems begin with finding an instantaneous rate of change at a point. It helps connect the graph of a function, the slope of a secant line, and the slope of a tangent line. The goal is to make the derivative definition clear and usable.

Key Facts

The slope of the secant line through $x=a$ and $x=a+h$ is $\frac{f(a+h)-f(a)}{h}$ , where $h \neq 0$ .
The derivative of $f$ at $x=a$ is $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ if the limit exists.
An equivalent derivative definition is $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$ .
The tangent line to $y=f(x)$ at $x=a$ has slope $f'(a)$ and passes through $(a,f(a))$ .
The tangent line equation is $y-f(a)=f'(a)(x-a)$ .
If $f'(a)>0$ , the function is increasing near $x=a$ ; if $f'(a)<0$ , the function is decreasing near $x=a$ .
A derivative does not exist at $x=a$ if the limit of the difference quotient fails to exist, such as at a corner, cusp, vertical tangent, or discontinuity.
The units of $f'(a)$ are the units of $f(x)$ divided by the units of $x$ .

Vocabulary

Secant line: A secant line is a line that passes through two points on a graph, such as $(a,f(a))$ and $(a+h,f(a+h))$ .
Tangent line: A tangent line is the line that best matches the direction of a curve at one point and has slope $f'(a)$ .
Difference quotient: The difference quotient $\frac{f(a+h)-f(a)}{h}$ gives the average rate of change from $x=a$ to $x=a+h$ .
Derivative at a point: The derivative at a point is the instantaneous rate of change, defined by $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ .
Instantaneous rate of change: An instantaneous rate of change describes how fast a function is changing at a single input value.
Point-slope form: Point-slope form is $y-y_1=m(x-x_1)$ , which is useful for writing a tangent line when a point and slope are known.

Common Mistakes to Avoid

Using $\frac{f(a+h)-f(a)}{a+h-a}$ but not simplifying the denominator to $h$ is wrong because the derivative limit depends on the quotient $\frac{f(a+h)-f(a)}{h}$ before taking $h\to 0$ .
Substituting $h=0$ too early is wrong because $\frac{f(a+h)-f(a)}{h}$ is undefined at $h=0$ and must be simplified before evaluating the limit.
Finding $f'(x)$ but forgetting to evaluate at $x=a$ is wrong because the tangent line at $x=a$ needs the specific slope $f'(a)$ .
Using $f(a)$ as the slope of the tangent line is wrong because $f(a)$ is the height of the graph, while $f'(a)$ is the slope.
Writing the tangent line through the wrong point is wrong because the tangent line at $x=a$ must pass through $(a,f(a))$ , not through $(a,f'(a))$ .

Practice Questions

1 For $f(x)=x^2+3x$ , use $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ to find $f'(2)$ .
2 Find the equation of the tangent line to $f(x)=x^2-4x+1$ at $x=3$ .
3 Use the limit definition $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$ to find $f'(1)$ for $f(x)=\frac{1}{x}$ .
4 Explain why a function with a sharp corner at $x=a$ may fail to have a derivative at $x=a$ , even if the function is continuous there.

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