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 $f(x_0)=f^{\prime }(x_0)=f^{″}(x_0)=f^{‴}(x_0)=g(x_0)=g^{\prime }(x_0)=g^{″}(x_0)=g^{‴}(x_0)=1$. 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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