# Category: Chain rule derivative definition

The chain rule is a formula to calculate the derivative of a composition of functions. Once you have a grasp of the basic idea behind the chain rule, the next step is to try your hand at some examples. Since the functions were linear, this example was trivial. Solution : This problem is a chain rule problem in disguise. This problem is the same as the previous example in disguise.

Solution : Again, we must use the chain rule.

It's OK if we use different notation for the functions or the inputs of the functions. Typically, when using the chain rule, we won't bother with the extra steps of defining the component functions. For additional examples, see the chain rule page from the Calculus Refresher.

Home Threads Index About. Simple examples of using the chain rule. Thread navigation MathFall Previous: The idea of the chain rule Next: Problem set: Quotient rule and chain rule MathSpring 21 Previous: The idea of the chain rule Next: Worksheet: Quotient rule and chain rule Similar pages A refresher on the chain rule The idea of the chain rule A refresher on the quotient rule A refresher on the product rule The quotient rule for differentiation Introduction to the multivariable chain rule Multivariable chain rule examples Special cases of the multivariable chain rule The idea of the derivative of a function Derivatives of polynomials More similar pages.

See also The idea of the chain rule A refresher on the chain rule.In calculusthe chain rule is a formula to compute the derivative of a composite function. The chain rule may also be rewritten in Leibniz's notation in the following way.

If a variable z depends on the variable ywhich itself depends on the variable x i. In which case, the chain rule states that:.

Intuitively, the chain rule states that knowing the instantaneous rate of change of z relative to y and that of y relative to x allows one to calculate the instantaneous rate of change of z relative to x. As put by George F. In integrationthe counterpart to the chain rule is the substitution rule. The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He first mentioned it in a memoir with a sign error in the calculation. The common notation of chain rule is due to Leibniz.

The chain rule does not appear in any of Leonhard Euler 's analysis books, even though they were written over a hundred years after Leibniz's discovery. The simplest form of the chain rule is for real-valued functions of one real variable.

It states that if g is a function that is differentiable at a point c i.

It may be possible to apply the chain rule even when there are no formulas for the functions which are being differentiated. This can happen when the derivatives are measured directly. Suppose that a car is driving up a tall mountain. The car's speedometer measures its speed directly. If the grade is known, then the rate of ascent can be calculated using trigonometry.

Suppose that the car is ascending at 2. Standard models for the Earth's atmosphere imply that the temperature drops about 6. To find the temperature drop per hour, we can apply the chain rule. Let the function g t be the altitude of the car at time tand let the function f h be the temperature h kilometers above sea level. The chain rule states that the derivative of the composite function is the product of the derivative of f and the derivative of g. A more accurate description of how the temperature near the car varies over time would require an accurate model of how the temperature varies at different altitudes.

This model may not have a constant derivative. The chain rule can be applied to composites of more than two functions. Applying the chain rule in this manner would yield:. This is the same as what was computed above. In this case, define. Then the chain rule takes the form. The chain rule can be used to derive some well-known differentiation rules.If you're seeing this message, it means we're having trouble loading external resources on our website.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Common chain rule misunderstandings.

Identifying composite functions. Practice: Identify composite functions. Practice: Chain rule intro. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter.

Video transcript What I want to do in this video is start with the abstract-- actually, let me call it formula for the chain rule, and then learn to apply it in the concrete setting. So let's start off with some function, some expression that could be expressed as the composition of two functions.

So it can be expressed as f of g of x. So it's a function that can be expressed as a composition or expression that can be expressed as a composition of two functions.

## Math Insight

Let me get that same color. I want the colors to be accurate. And my goal is to take the derivative of this business, the derivative with respect to x. And what the chain rule tells us is that this is going to be equal to the derivative of the outer function with respect to the inner function. And we can write that as f prime of not x, but f prime of g of x, of the inner function. Now this might seem all very abstract and math-y.

How do you actually apply it? Well, let's try it with a real example. Let's say we were trying to take the derivative of the square root of 3x squared minus x.

So how could we define an f and a g so this really is the composition of f of x and g of x?

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Well, we could define f of x. If we defined f of x as being equal to the square root of x, and if we defined g of x as being equal to 3x squared minus x, then what is f of g of x? Well, f of g of x is going to be equal to-- I'm going to try to keep all the colors accurate, hopefully it'll help for the understanding. So this thing right over here is exactly f of g of x if we define f of x in this way and g of x in this way.In mathematicsthe derivative of a function of a real variable measures the sensitivity to change of the function value output value with respect to a change in its argument input value.

Derivatives are a fundamental tool of calculus.

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For example, the derivative of the position of a moving object with respect to time is the object's velocity : this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value.

For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is after an appropriate translation the best linear approximation to the graph of the original function.

The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration.

Differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative. It is called the derivative of f with respect to x. If x and y are real numbersand if the graph of f is plotted against xthe derivative is the slope of this graph at each point. The simplest case, apart from the trivial case of a constant functionis when y is a linear function of xmeaning that the graph of y is a line. If the function f is not linear i.

Two distinct notations are commonly used for the derivative, one deriving from Gottfried Wilhelm Leibniz and the other from Joseph Louis Lagrange. A third notation, first used by Isaac Newtonis sometimes seen in physics. In Leibniz's notationan infinitesimal change in x is denoted by dxand the derivative of y with respect to x is written.

The above expression is read as "the derivative of y with respect to x ", " dy by dx ", or " dy over dx ". The oral form " dy dx " is often used conversationally, although it may lead to confusion. Lagrange's notation is sometimes incorrectly attributed to Newton. Newton's notation for differentiation also called the dot notation for differentiation places a dot over the dependent variable. That is, if y is a function of tthen the derivative of y with respect to t is.

Newton's notation is generally used when the independent variable denotes time. The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers.

Let f be a real valued function defined in an open neighborhood of a real number a. In classical geometry, the tangent line to the graph of the function f at a was the unique line through the point af a that did not meet the graph of f transversallymeaning that the line did not pass straight through the graph.

The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at af a. These lines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values in absolute value of h will, in general, give better approximations.

The slope m of the secant line is the difference between the y values of these points divided by the difference between the x values, that is. This expression is Newton 's difference quotient. Passing from an approximation to an exact answer is done using a limit.Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I have seen many proofs using the normal derivative definition, but I was wondering if the same result could be achieved using the sequential definition which states:.

Sign up to join this community. The best answers are voted up and rise to the top. Proving the chain rule using the sequential definition of derivatives Ask Question. Asked 3 years, 9 months ago. Active 1 year, 10 months ago. Viewed times. Lee David Chung Lin 6, 8 8 gold badges 21 21 silver badges 45 45 bronze badges.

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Question feed. Mathematics Stack Exchange works best with JavaScript enabled.In this chapter we will start looking at the next major topic in a calculus class, derivatives.

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This chapter is devoted almost exclusively to finding derivatives. We will be looking at one application of them in this chapter. We will be leaving most of the applications of derivatives to the next chapter. The Definition of the Derivative — In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function.

Interpretation of the Derivative — In this section we give several of the more important interpretations of the derivative. We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function.

Differentiation Formulas — In this section we give most of the general derivative formulas and properties used when taking the derivative of a function.

Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers. Product and Quotient Rule — In this section we will give two of the more important formulas for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate. Derivatives of Trig Functions — In this section we will discuss differentiating trig functions. Derivatives of Exponential and Logarithm Functions — In this section we derive the formulas for the derivatives of the exponential and logarithm functions.

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Derivatives of Inverse Trig Functions — In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent. Derivatives of Hyperbolic Functions — In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions.

We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine. Chain Rule — In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! Implicit Differentiation — In this section we will discuss implicit differentiation.

Not every function can be explicitly written in terms of the independent variable, e. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates the next section. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one or more quantities in the problem.

This is often one of the more difficult sections for students.

### Chain Rule

We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work.

Higher Order Derivatives — In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives.If you're seeing this message, it means we're having trouble loading external resources on our website.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Chain rule. Common chain rule misunderstandings. Identifying composite functions.

Practice: Identify composite functions. Practice: Chain rule intro. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript - [Instructor] What we're going to go over in this video is one of the core principles in calculus, and you're going to use it any time you take the derivative, anything even reasonably complex.

And it's called the chain rule. And when you're first exposed to it, it can seem a little daunting and a little bit convoluted.

But as you see more and more examples, it'll start to make sense, and hopefully it'd even start to seem a little bit simple and intuitive over time. So let's say that I had a function. Let's say I have a function h of x, and it is equal to, just for example, let's say it's equal to sine of x, let's say it's equal to sine of x squared.

Now, I could've written that, I could've written it like this, sine squared of x, but it'll be a little bit clearer using that type of notation.

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So let me make it so I have h of x. And what I'm curious about is what is h prime of x? So I want to know h prime of x, which another way of writing it is the derivative of h with respect to x.

These are just different notations. And to do this, I'm going to use the chain rule. I'm going to use the chain rule, and the chain rule comes into play every time, any time your function can be used as a composition of more than one function. And as that might not seem obvious right now, but it will hopefully, maybe by the end of this video or the next one. Now, what I want to do is a little bit of a thought experiment, a little bit of a thought experiment.

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