TsAmE said:Homework Statement
Sketch the graph of x ^ (4/9) * e ^ (-x)
Homework Equations
None.
The Attempt at a Solution
My y' = -x ^ (4/9) * e ^ (-x) ( 1 - 4/9x ^ 1/9). I keep on getting a reaaally long derivative for y'' and thus cannot place it on my sign table. Could someone please show me the correct steps in order to get a simplified y''?
Simplifying derivatives when sketching graphs is important because it helps us to understand the overall behavior of a function and how it changes at different points. It also allows us to identify key features, such as maximum and minimum points, and inflection points.
Some common techniques for simplifying derivatives include using the power rule, product rule, quotient rule, and chain rule. It is also helpful to know basic algebraic simplification techniques, such as factoring and canceling common terms.
By simplifying derivatives, we can find the critical points of a function, which are points where the derivative is equal to zero or undefined. These critical points can tell us where a graph has maximum or minimum values, as well as inflection points where the concavity of the graph changes.
One common mistake is to forget to apply the chain rule when differentiating composite functions. It is also important to be careful with algebraic simplification and to double-check your work for any potential mistakes or errors.
Practice is key! Make sure to fully understand the basic derivative rules and how they apply to different types of functions. Additionally, it can be helpful to work on a variety of examples and to seek out additional resources, such as textbooks or online tutorials, for extra practice and guidance.