Quiz:Product rule for higher derivatives

ORIGINAL FULL PAGE: Product rule for higher derivatives
STUDY THE TOPIC AT MULTIPLE LEVELS:
ALSO CHECK OUT: Quiz (multiple choice questions to test your understanding) |

For a quiz that tests all the differentiation rules together, see Quiz:Differentiation rules.

Practical

Suppose $f$ and $g$ are continuous functions at $x_{0}$. Suppose $f(x_{0})=f'(x_{0})=f''(x_{0})=f'''(x_{0})=g(x_{0})=g'(x_{0})=g''(x_{0})=g'''(x_{0})=1$. What is $(f\cdot g)'''(x_{0})$?

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Significance

Qualitative and existential significance

Suppose $f$ and $g$ are continuous functions at $x_{0}$. Suppose we know that $f$ is 5 times differentiable at $x_{0}$ and $g$ is 3 times differentiable at $x_{0}$. What is the maximum number of times we can be sure (from this information) that $f\cdot g$ is differentiable at $x_{0}$?

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