Page:Calculus Made Easy.pdf/35

 these other variables are called the independent variables because their value is not determined from the value assumed by the function. For example, if $$u=x^2\sin \theta$$, $$x$$ and $$\theta$$ are the independent variables, and $$u$$ is the dependent variable.

Sometimes the exact relation between several quantities $$x$$, $$y$$, $$z$$ either is not known or it is not convenient to state it; it is only known, or convenient to state, that there is some sort of relation between these variables, so that one cannot alter either $$x$$ or $$y$$ or $$z$$ singly without affecting the other quantities; the existence of a function in $$x$$, $$y$$, $$z$$ is then indicated by the notation $$F(x,y,z)$$ (implicit function) or by $$x=F(y,z)$$, $$y=F(x,z)$$ or $$z=F(x,y)$$ (explicit function). Sometimes the letter $$f$$ or $$\phi$$ is used instead of $$F$$, so that $$y=F(x)$$, $$y=f(x)$$ and $$y=\phi (x)$$ all mean the same thing, namely, that the value of $$y$$ depends on the value of $$x$$ in some way which is not stated.

We call the ratio $$\frac{dy}{dx}$$, “the differential coefficient of $$y$$ with respect to $$x$$.” It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy. Instead of being frightened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names; and, having relieved our minds, will go on to the simple thing itself, namely the ratio $$\frac{dy}{dx}$$.