Category:Examples of Use of Quotient Rule for Derivatives
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This category contains examples of use of Quotient Rule for Derivatives.
Let $\map j x, \map k x$ be real functions defined on the open interval $I$.
Let $\xi \in I$ be a point in $I$ at which both $j$ and $k$ are differentiable.
Define the real function $f$ on $I$ by:
- $\ds \map f x = \begin{cases}
\dfrac {\map j x} {\map k x} & : \map k x \ne 0 \\ 0 & : \text{otherwise} \end{cases}$
Then, if $\map k \xi \ne 0$, $f$ is differentiable at $\xi$, and furthermore:
- $\map {f'} \xi = \dfrac {\map {j'} \xi \map k \xi - \map j \xi \map {k'} \xi} {\paren {\map k \xi}^2}$
It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then:
- $\ds \forall x \in I: \map k x \ne 0 \implies \map {f'} x = \frac {\map {j'} x \map k x - \map j x \map {k'} x} {\paren {\map k x}^2}$
Pages in category "Examples of Use of Quotient Rule for Derivatives"
The following 6 pages are in this category, out of 6 total.
Q
- Quotient Rule for Derivatives/Examples
- Quotient Rule for Derivatives/Examples/(x-1)(2x-1) over x-2
- Quotient Rule for Derivatives/Examples/Exponential of x over x
- Quotient Rule for Derivatives/Examples/x over Cosine of x
- Quotient Rule for Derivatives/Examples/x over x+1
- Quotient Rule for Derivatives/Examples/x+1 over x-1