Product Rule: Proof, Higher Derivatives, and Extensions

All flashcards

Product rule

For differentiable functions $u$ and $v$, $(uv)'=u'v+uv'$.

Product-rule proof by limits

Add and subtract $f(x)g(x+\Delta x)$ to split the product's difference quotient into two parts.

Second derivative of a product

For twice-differentiable $u$ and $v$, $(uv)''=u''v+2u'v'+uv''$.

General Leibniz rule

The $n$th derivative of $uv$ is $\sum_{k=0}^{n}\binom{n}{k}u^{(n-k)}v^{(k)}$.

Graded product rule for differential forms

For a $k$-form $\alpha$, $d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^k\alpha\wedge d\beta$.

Quiz questions

1. If $u$ and $v$ are differentiable, which formula gives the derivative of their product?Answer: $(uv)'=u'v+uv'$
2. In the limit proof for $h(x)=f(x)g(x)$, which expression is added and subtracted in the numerator to split the difference quotient?Answer: $f(x)g(x+\Delta x)$
3. For twice-differentiable functions $u$ and $v$, what is $(uv)''$?Answer: $u''v+2u'v'+uv''$
4. If differentiable vector functions $\mathbf f$ and $\mathbf g$ have a dot product, how is its derivative expressed?Answer: $(\mathbf f\cdot\mathbf g)'=\mathbf f'\cdot\mathbf g+\mathbf f\cdot\mathbf g'$
5. In an induction proof of the power rule, which expression results from differentiating $x^n x$ using the induction hypothesis $(x^n)'=nx^{n-1}$?Answer: $x(nx^{n-1})+x^n$