Differentials and Partial Derivatives the quotient of df and dx .Well, in some cases yes. For instance, if f x mx c m c are constants, then, y ( ) . D y = f x m x + ∆ ∆= ′ ∆ for all x ∈ and D x and hence equality in both ( ), and ( ). In this case changes in x and y are taking place along straight lines, in which case we have, change in change in x = ∆ ∆ = ′ df Thus in this case the derivative df dx is truly a quotient of df and , if we take df = ∆= and = ∆ .
This leads us to define the differential of f as follows: Definition . Let f a b :( , ) → be a differentiable function, for x a b ∈ ( , ) and the increment given to x , we define the differential of f by df ′ ( ) ∆ . ... ( ) First we note that if f x ( ) = , then by ( ) we get dx ′ ( ) ∆ ∆ which means that the differential dx = ∆ , which is the change in x -axis.
So the differential given by ( ) is same as df ′ ( ) Next we explore the differential for an arbitrary differentiable function y ( ) . Then ∆ f ( ) gives the change in output along the graph of ( ) and ′ f ( ) gives the slope of the tangent line at ( , ( )) x f x . Let dy or df denote the increment in f along the tangent line. Then by the above observation, we have dy ′ ( ) From the figure it is clear that ∆≈ ′ df and hence ′ f ( ) can be viewed approximately as the quotient of D f and D x .
So we may interpret df dx as the quotient of df and dx . Remark We know that derivative of