# Chain Rule with Leibniz Notation

• opus
In summary, the derivative of y=cos^3(πx) can be found by taking the derivative of y with respect to u, cos(u), and πx.
opus
Gold Member

## Homework Statement

Find the derivative of ##y=cos^3(πx)##
*Must be in Leibniz notation

## The Attempt at a Solution

(i) $$Let~ w=y^3 , y=cos(u), u=πx$$

(ii) $$\frac{dw}{dy} = 3y^2,~ \frac{dy}{du} = -sin(u),~ \frac{du}{dx}=π$$

(iii) By the Chain Rule,
$$\frac{dw}{dx} = \frac{dw}{dy}⋅\frac{dy}{du}⋅\frac{du}{dx}$$

(iv) $$= 3cos^2(πx)⋅-sin(πx)⋅π$$

(v) $$=-3πcos^2(πx)sin(πx)$$

I'm not sure if this is correct. We're told to use Leibniz notation, but were taught the Chain Rule in "prime notation", so I just put things together in what made some sort of sense. So I'd like to know if this is correct, and if I'm using this notation correctly.

I looks just fine to me.

opus
Im surprised to hear that. Is there anything special about this notation? Why choose this over using primes? Or maybe its just a practice thing?

opus said:
Im surprised to hear that. Is there anything special about this notation? Why choose this over using primes? Or maybe its just a practice thing?
The two notations aren't in a competition of better and worse. They both serve their own purposes. The Leibniz notation has the advantage that the variable is noted, which in case of more than one variable, or here, in case of change the variable makes sense. You cannot see this from the prime notation. Note that ##\dfrac{df(x)}{dx}=f'(x)## but ##\dfrac{df(x)}{dy}=0##. It also makes sense for future applications in differential geometry, as the ##dx## work as basis vectors there.

The prime notation is shorter and often does the job when it is clear what the variable, i.e. direction is, along which we differentiate. If there is only one, fine, but if there are more than one, it becomes necessary to distinguish them.

It also has another advantage which is usually hidden behind the notation. If you write ##f'(x)## then you normally mean the slope at ##x##. Let's call this point ##x=a##. Then what you really mean is ##f'(a)## which is ##\left. \dfrac{d}{dx}\right|_{x=a}f(x) = f'(a)##. However, we may also consider the function ##a \mapsto f'(a)## i.e. consider the dependency point → slope at this point. So the Leibniz notation forces you to be clear which role ##x## is actually playing.

opus
Ok that makes sense. I have to say that I didn't like it at first, but after finishing my homework I like it more for some of the same reasons that you stated. It seems more specific in that it specifies exactly what I'm working with. Thanks guys.

## What is the Chain Rule with Leibniz Notation?

The Chain Rule with Leibniz Notation is a mathematical concept that is used to find the derivative of a composite function. It allows us to calculate the rate of change of a function that is composed of two or more functions.

## Why is the Chain Rule with Leibniz Notation important?

The Chain Rule with Leibniz Notation is important because it is a fundamental tool in calculus that is used to solve problems related to rates of change. It is also crucial in understanding more complex mathematical concepts such as implicit differentiation and related rates.

## How does the Chain Rule with Leibniz Notation work?

The Chain Rule with Leibniz Notation states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. In other words, it allows us to break down a complex function into smaller, more manageable parts.

## What is the formula for the Chain Rule with Leibniz Notation?

The formula for the Chain Rule with Leibniz Notation is: dy/dx = dy/du * du/dx, where u is the inner function and y is the outer function. This formula can also be written in Leibniz notation as d(u(y))/dx = d(u)/dy * dy/dx.

## Can the Chain Rule with Leibniz Notation be applied to any function?

Yes, the Chain Rule with Leibniz Notation can be applied to any function, as long as it is composed of two or more functions. However, it may become more complex for functions with multiple layers of composition or for functions with trigonometric or exponential components.

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