MAT 101 Lecture Notes - Differentiable Function, Dependent And Independent Variables, Product Rule
Document Summary
4. 1 definition of function derivative changes with respect to the change in the independent variable. Differentiation is a method to compute the rate at which a dependent variable (cid:1877) called the derivative of (cid:1877) with respect to (cid:1876). The derivative of the function (cid:1858)((cid:1876)) with respect to the variable (cid:1876) is the function (cid:1858)(cid:4593) whose value at (cid:1876) is. Definition of derivative function (cid:1858)(cid:4593)((cid:1876))=lim(cid:3035) (cid:2868)(cid:1858)((cid:1876)+ ) (cid:1858)((cid:1876)) provided the limit exists notations for the derivative are. There are many ways to denote the derivative of a function (cid:1877)=(cid:1858)((cid:1876)) where the independent variable is (cid:1876) and the dependent variable is (cid:1877). Example 4. 1 show that (cid:1877)=|(cid:1876)| is differentiable on ( ,0) and (0, ) but has no derivative at the origin (cid:1876)=0. There can be no derivative at the origin because the one-sided derivatives differ there: 4. 3. 1 constant rule (1) (cid:1835)(cid:1858) (cid:1858)((cid:1876))=(cid:1876)(cid:2872) (cid:1858)(cid:4593)((cid:1876))=4 (cid:1876)(cid:2871). (2) (cid:1835)(cid:1858) (cid:1858)((cid:1876))=(cid:1876)(cid:2876) (cid:1858)(cid:4593)((cid:1876))=8 (cid:1876)(cid:2875).
