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5 days ago · A derivation is a sequence of steps, logical or computational, from one result to another. The word derivation comes from the word "derive." "Derivation" can also refer to a particular type of operator used to define a derivation algebra on a ring or algebra. In particular, let A be a Banach algebra and X be a Banach A-bimodule.
See how we define the derivative using limits, and learn to find derivatives quickly with the very useful power, product, and quotient rules. The derivative of a function describes the function's instantaneous rate of change at a certain point - it gives us the slope of the line tangent to the function's graph at that point.
Aug 10, 2020 · The noun for what we are finding is “the derivative “, which basically means “a related function we have derived from the given function”. But the verb we use for that process is not “to derive”, but “to differentiate “, which comes from the “ difference quotient ” on which the derivative is based.
Define \[u(x)=\left\{\begin{array}{ll} \frac{f(x)-f(a)}{x-a}-f^{\prime}(a), & x \in G \backslash\{a\} \\ 0, & x=a \end{array}\right .\] Since \(f\) is differentiable at \(a\), we have \[\lim _{x \rightarrow a} u(x)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}-f^{\prime}(a)=f^{\prime}(a)-f^{\prime}(a)=0 .\]
Sep 28, 2023 · The limit definition of the derivative produces a value for each x at which the derivative is defined, and this leads to a new function whose formula is y = f' (x). Hence we talk both about a given …
We can equivalently define the derivative \(f'(a)\) by the limit \begin{gather*} f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}. \end{gather*} To see that these two definitions are the same, we set \(x=a+h\) and then the limit as \(h\) goes to \(0\) is equivalent to the limit as \(x\) goes to \(a\text{.}\)
