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### How to Solve Limits with Basic Algebra

When substitution doesn’t work in the original limit function — usually because of a hole in the function — you can often use some algebra to manipulate the function until substitution does work [more…]

### How to Solve Limits at Infinity with a Calculator

Solving for limits at infinity is easy to do when you use a calculator. For example, enter the below function in your calculator's graphing mode: [more…]

### How to Solve Limits at Infinity by Using Algebra

Yes, you can solve a limit at infinity using a calculator, but all things being equal, it’s better to solve the problem algebraically, because then you have a mathematically airtight answer. For example [more…]

### How to Find the Derivative of a Line

The *derivative* is just a fancy calculus term for a simple idea that you probably know from algebra — slope. *S**lope* is the fancy algebra term for steepness. And [more…]

### How to Find the Derivative of a Curve

Calculus is the mathematics of change — so you need to know how to find the derivative of a *parabol**a**,*which is a curve with a constantly changing slope. [more…]

### How to Differentiate the Trigonometric Functions

You should memorize the derivatives of the six trig functions. Make sure you memorize the first two in the following list — they’re a snap. If you’re good at rote memorization, memorize the last four as [more…]

### How to Differentiate Exponential and Logarithmic Functions

Differentiating exponential and logarithmic functions involves special rules. No worries — once you memorize a couple of rules, differentiating these functions is a piece of cake. [more…]

### How to Find Derivatives Using the Product and Quotient Rules

There are special rules for finding the derivative of the product of two functions or the quotient of two functions; these are the product rule and the quotient rule, respectively. [more…]

### How to Differentiate Implicitly

In most differentiation problems, *y* is written *explicitly* as a function of *x*, which just means that the equation is solved for *y*; in other words, *y* is by itself on one side of the equation. Sometimes, [more…]

### How to Use Logarithmic Differentiation

For differentiating certain functions, logarithmic differentiation is a great shortcut. It spares you the headache of using the product rule or of multiplying the whole thing out and then differentiating [more…]

### How to Differentiate Inverse Functions

There’s a difficult-looking formula involving the derivatives of inverse functions, but before you get to that, look at the following figure, which nicely sums up the whole idea. [more…]

### How to Find High-Order Derivatives

Finding a second, third, fourth, or higher derivative is incredibly simple. The second derivative of a function is just the derivative of its first derivative. The third derivative is the derivative of [more…]

### How to Find a Function’s Absolute Extrema over Its Entire Domain

A function’s *absolute max* and *absolute min* over its *entire domain* are the single highest and single lowest values of the function anywhere it’s defined. A function can have an absolute max or min or both [more…]

### The Mean Value Theorem

You don’t need the mean value theorem for much, but it’s a famous theorem — one of the two or three most important in all of calculus — so you really should learn it. Fortunately, it’s very simple. [more…]

### How to Use Differentiation to Calculate the Maximum Volume of a Box

One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function. In the following example, you calculate the maximum volume of a box that has no top and [more…]

### How to Use Differentiation to Calculate the Maximum Area of a Corral

Finding the maximum or minimum value of a real-world function is one of the most practical uses of differentiation. For example, you might need to find the maximum area of a corral, given a certain length [more…]

### Related Rates: the Expanding Balloon Problem

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

### Related Rates: the Trough of Swill Problem

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

### Related Rates: Two Cars at a Crossroads

Say you’re filling up your swimming pool and you know how fast water is coming out of your hose, and you want to calculate how fast the water level in the pool is rising. You know one rate [more…]

### How to Find the Tangent Lines of a Parabola that Pass through a Certain Point

Ever want to determine the location of a line through a given point that’s tangent to a given curve? Of course you have! Here’s how you do it.

Determine the points of tangency of the lines through the point [more…]

### How to Find a Normal Line to a Curve

A line *normal* to a curve at a given point is the line perpendicular to the line that’s tangent at that same point. Find the points of perpendicularity for all normal lines to the parabola [more…]

### How to Determine Marginal Cost, Marginal Revenue, and Marginal Profit in Economics

Marginal cost, marginal revenue, and marginal profit all involve how much a function goes up (or down) as you go over 1 to the right — this is very similar to the way linear approximation works. [more…]

### How to Approximate Area with Left Rectangles

You can approximate the area under a curve by summing up “left” rectangles. For example, say you want the area under the curve *f* (*x*) = *x*^{2} + 1 from 0 to 3. The shaded area of the graph on the left side [more…]

### How to Approximate Area with Midpoint Rectangles

A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangle's top side*.* A midpoint sum is a much better estimate of area than either a left-rectangle [more…]

### How to Approximate Area with the Trapezoid Rule

With the trapezoid rule, instead of approximating area by using rectangles (as you do with the left, right, and midpoint rectangle methods), you approximate area with — can you guess? — trapezoids. [more…]