If the original function is the product of three functions, then its derivative is the sum of three products. If the original function is the product of four functions, then its derivative is the sum of four products.

Read MoreWhy use subscripts instead of different variables? Well, the variables in science often represent something specific. For example, P stands for pressure and in some equations you’ll have more than one pressure represented. In that case we use subscripts: P_1, P_2, P_3, and so on.

Read MoreThe orthogonal trajectories to a family of curves are the curves that intersect each member of the family at a perfectly perpendicular angle. So given a family of curves, you can change the value of the constant in the equation that models the family, to create a family of many curves, and then sketch the family in the plane.

Read MoreIf two planes intersect each other, the intersection will always be a line. The vector equation for the line of intersection is calculated using a point on the line and the cross product of the normal vectors of the two planes.

Read MoreMost integrals need some work before you can even begin the integration. They have to be transformed or manipulated in order to reduce the function’s form into some simpler form. U-substitution is the simplest tool we have to transform integrals.

Read MoreThe definition of the derivative, also called the “difference quotient”, is a tool we use to find derivatives “the long way”, before we learn all the shortcuts later that let us find them “the fast way”.

Read MoreTo solve simple equations, start by thinking about what’s happening to the variable.

Read MoreMixing problems are an application of separable differential equations. They’re word problems that require us to create a separable differential equation based on the concentration of a substance in a tank.

Read MoreTo say whether the planes are parallel, we’ll set up our ratio inequality using the direction numbers from their normal vectors.

Read MoreThe fundamental theorem of calculus (FTC) is the formula that relates the derivative to the integral and provides us with a method for evaluating definite integrals.

Read MoreThe squeeze theorem allows us to find the limit of a function at a particular point, even when the function is undefined at that point.

Read MoreIn this lesson we’ll work with both positive and negative fractional exponents.

Read MoreThink of an equation as a balance scale that must always be balanced. What you do to one side of an equation you must do to the other in order for it to remain balanced.

Read MoreWe need to change the current equation so that it's in terms of a new variable u and its derivative u'.

Read MoreTo find the radius, it’s important that we take the square root of the right-hand side, and not just the full value from the right

Read MoreEvaluating a definite integral means finding the area enclosed by the graph of the function and the x-axis, over the given interval [a,b].

Read More“Limits at infinity” sounds a little mysterious, and it can be difficult to imagine the concept when we first hear this term. But let’s start by remembering that limits can be defined as the restrictions on the continuity of a function.

Read MoreRemember that any number can be written as itself divided by 1. For example, 3 is the same as 3/1. Also remember that the top part of a fraction is called the numerator and the bottom part of a fraction is called the denominator.

Read MoreThe distributive property can be used even when there are two sets of parentheses with two terms each. It’s called binomial multiplication (remember that a bicycle has two wheels and a binomial has two terms).

Read MoreWhat you want to do is create a field of equally spaced coordinate points, and then evaluate the derivative at each of those coordinate points. Since the derivative is the same thing as the slope of the tangent line, finding the derivative at a particular point is like finding the slope of the tangent line there, which of course is an approximation of the slope of the actual function.

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