Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

The differential of a differentiable function u(x).

Short answer

If u(x) is a differentiable function of x, then differential du is equal to$du=u\text{'}\left(x\right)dx$

Step-by-step solution

Step 1. Given Information

The given statement is the differential of a differentiable function u(x).

Step 2. Explanation

If u(x) is a differentiable function of x, then differential du is equal to $du=u\text{'}\left(x\right)dx$

For example, consider the integral
localid="1649528992394" $$\begin{gathered} \cos x^2\left(2x\right)dx=2x\cos x^{2}dx \\ \mathrm{Let}u\left(x\right)=x^2\Rightarrow \frac{du}{dx}=2x \\ \Rightarrow du=2xdx \end{gathered}$$