# Derivatives from the Rules of Differentiation

The product rule. A constant times a function rule. Derivatives of functions, examples. The quotient rule. The derivative of the tangent function, use of the quotient rule. The derivative of the cotangent function, use of the quotient rule. Maximums and minimums We use derivatives to help locate extrema.

Concepts of graphing functions. Two young mathematicians discuss how to sketch the graphs of functions.

## Differentiation: definition and basic derivative rules | Khan Academy

Concepts of graphing functions We use the language of calculus to describe graphs of functions. Computations for graphing functions. Wanted: graphing procedure Two young mathematicians discuss how to sketch the graphs of functions. Computations for graphing functions We will give some general guidelines for sketching the plot of a function.

Mean Value Theorem. The Extreme Value Theorem We examine a fact about continuous functions. The Mean Value Theorem Here we see a key theorem of calculus. Replacing curves with lines Two young mathematicians discuss linear approximation. Explanation of the product and chain rules We give explanation for the product rule and chain rule. A mysterious formula Two young mathematicians discuss optimization from an abstract point of view.

Basic optimization Now we put our optimization skills to work. Applied optimization. Volumes of aluminum cans Two young mathematicians discuss optimizing aluminum cans. Applied optimization Now we put our optimization skills to work. A limitless dialogue Two young mathematicians consider a way to compute limits using derivatives. Basic antiderivatives We introduce antiderivatives. Falling objects We study a special type of differential equation.

Approximating the area under a curve. What is area? Two young mathematicians discuss the idea of area. Introduction to sigma notation We introduce sigma notation. Approximating area with rectangles We introduce the basic idea of using rectangles to approximate the area under a curve. Definite integrals.

Computing areas Two young mathematicians discuss cutting up areas. The definite integral Definite integrals compute net area. Antiderivatives and area. Meaning of multiplication A dialogue where students discuss multiplication. Relating velocity, displacement, antiderivatives and areas We give an alternative interpretation of the definite integral and make a connection between areas and antiderivatives.

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## 6. Derivatives of Products and Quotients

First Fundamental Theorem of Calculus. Two young mathematicians discuss what calculus is all about. The First Fundamental Theorem of Calculus The rate that accumulated area under a curve grows is described identically by that curve. Second Fundamental Theorem of Calculus.

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A secret of the definite integral Two young mathematicians discuss what calculus is all about. The Second Fundamental Theorem of Calculus The accumulation of a rate is given by the change in the amount. Applications of integrals. What could it represent?

Two young mathematicians discuss whether integrals are defined properly. Applications of integrals We give more contexts to understand integrals. The idea of substitution.

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Geometry and substitution Two students consider substitution geometrically. The idea of substitution We learn a new technique, called substitution, to help us solve problems involving integration. Working with substitution. Integrals are puzzles! Two young mathematicians discuss how tricky integrals are puzzles. Working with substitution We explore more difficult problems involving substitution. The Work-Energy Theorem Substitution is given a physical meaning.

We derive the constant rule, power rule, and sum rule. There are several different notations for the derivative. The constant rule Given a constant , From the limit definition of the derivative, write. What is equal to? Binomial Theorem If is a nonnegative integer, then where. Expand using the Binomial Theorem.

If u and v are two functions of x , then the derivative of the product uv is given by Where does this formula come from? Like all the differentiation formulas we meet, it is based on derivative from first principles. In Example 1 and Example 2, it appears that it would be easier to just expand out the brackets first, then differentiate the result.

It is not possible to multiply this expression term-by-term, so we need a method to differentiate products of such functions. Go to the differentiation applet to explore Examples 3 and 4 and see what we've found. Derivative graphs interactive. Derivative of square root of sine x by first principles.

Can we find the derivative of all functions? How to differentiate? Rate of change by Rismiya [Solved! More info: Calculus videos. Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight.