Decomposing Derivatives

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After completing this section, students should be able to do the following.

Calculate derivatives term by term using sum/difference rule.
Factor constant multiples out of derivatives as part of the differentiation process.
State the polynomial rule.
Properly use the polynomial rule while taking a derivative.
Apply the polynomial rule to terms with irrational and/or negative powers.
State the product rule for derivatives.
Apply the product rule to decompose the product of functions when taking derivatives.
State the quotient rule for derivatives.
Apply the quotient rule to compute derivatives of quotients of functions.
State the chain rule.
Apply the chain rule properly while taking a derivative.
Explain the geometric interpretation of the Chain Rule as a process of magnifying rates of change.
Apply chain rule to functions with multiple layers of composition.
Take derivatives of implicitly defined functions.
Identify when an expression requires an implicit differentiation.
Solve for the derivative (explicitly) in an implicitly defined function.