![]() The point of this example is that these $dx$ and $dy$ terms are quite flexible: they really do refer to a change in a variable and can directly translate physical meaning into mathematical meaning. Which really means "break up the path into small pieces and sum a quantity over each piece, taking into account both how $x$ changed and how $y$ changed". If, however, you were looking at the derivative with respect to $y$, then the gradient function would tell you what the gradient is for each $y$-value.įrom what I understand, $\fracF_1(x,y)\,dx + F_2(x,y)\,dy$$ For example, when $x=5$, the gradient is $10$. I am not completely clear on what "with respect to $x$" means, but I think it means that the derivative is telling you what the rate of change for each value of $x$ is. B.3 Problem Sheet 3 B.4 Problem Sheet 4 B.5 Problem Sheet 5 B.6 Problem. I have heard spoken aloud as "the rate of change of y of $x^2+5$ with respect to $x$ is $2x$". ![]() Part of this bewilderment stems from the notation (and the language used to describe the notation). While I understand the techniques for differentiation and integration, I still feel as if I don't understand why they work. So this idea, this is known as sometimes differential notation, Leibnizs notation, is instead of just change in y over change in x, super small changes in y for a super small change in x, especially as the change in x approaches zero, and as you will see, that is how we will calculate the derivative. Different inputs will mean the sum converges to different answers. ![]() ![]() The two most popular types are Prime notation (also called Lagrange notation) and Leibniz notation. Notation 3 We write C for the set of all complex numbers. I have only recently began studying calculus at school, so a non-technical answer would be greatly appreciated. There are a few different ways to write a derivative. ![]()
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