So, if we leave r = p / q {displaystyle r=p/q}, we can conclude that d d x x r = r x r − 1 {displaystyle {frac {d}{dx}}x^{r}=rx^{r-1}} if {displaystyle r} is a rational number. No wonder that, according to Wikipedia, it takes into account one of the most important rules in global differentiation. If f ( x ) = e x {displaystyle f(x)=e^{x}} , then ln ( f ( x ) ) = x {displaystyle ln(f(x))=x} , where ln {displaystyle ln } is the natural logarithm function, the inverse function of the exponential function, as Euler shows. [3] Since the last two functions are the same for all values of x > 0 {displaystyle x>0}, their derivatives are equal even if a derivative exists, so we can be derived by the chain rule for any real number r ≠ − 1 {displaystyle rneq -1} by applying the fundamental theorem of infinitesimal calculus to the power rule for differentiation. The power rule only works for functions incremented to a power, such as x^3, x^4, (x+2)^5 or sqrt(x), etc. Performance is not a variable, it is a constant. If the power is a variable, like e^x, 2^x, let`s call it an exponential function, and you can`t use the power rule to distinguish it. Think of the definition of the derivative, like f (x + h) — f (x) everywhere in h when h goes to zero, and look what happens for a function like x^2, x^3, x^4 (why the derivative of x^n * x^(n-1)? What`s happening is interesting to observe), then do the same for e^x, 2^x and see how it`s fundamentally different. Hope this helps! Use the power rule to distinguish each power function. Since (x) was for itself, its derivative is (1x^0). Normally, however, this is not announced. Remember that everything (except zero) is zero power 1.
So, (10left(x^0right) = 10(1) = 10). Therefore, we can write the final answer as follows: If we apply the rule for negative exponents, we can rewrite this function as follows: But did you know that we can apply the power rule and its conclusions to functions other than polynomials, such as functions that contain negative or rational exponents? In calculus, the power rule is used to distinguish functions of the form f ( x ) = x r {displaystyle f(x)=x^{r}} if r {displaystyle r} is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series because it relates a series of powers to the derivatives of a function. This helps us find the rate of change of performance features more «complicated». For example, let`s use the power rule to find the derivative of x^2. All we have to do is move the exponent forward and then decrease the exponent by 1. Let`s take a closer look at each of these rules of differentiation as we learn to apply our general rule of power. If you`re just starting to study calculus, just ask yourself: Is the exponent just a constant? If the answer is yes, then yes, you can use the power rule.
If not, no. We start by deriving a power function, $ds f(x)=x^n$. Here, $n$ is a number of any kind: integer, rational, positive, negative, even irrational, as in $ds x^pi$. We`ve already calculated a few simple examples, so the formula shouldn`t come as a complete surprise: $${dover dx}x^n = nx^{n-1}.$$ It`s not easy to show that this is true for every $$n. We will do some of the easiest cases now and discuss the rest later. If c {displaystyle c} is a positive integer, then no branching section is needed: one can define f ( 0 ) = 0 {displaystyle f(0)=0} , or positive integral complex powers by complex multiplication and show that f ′ ( z ) = c z c − 1 {displaystyle f`(z)=cz^{c-1}} for all complexes z {displaystyle z} , from the definition of the derivative and the binomial theorem. But how does the power rule apply to more complicated functions? For a negative integer n, let n = − m {displaystyle n=-m}, so that m is a positive integer. Using the reciprocal rule, d d x x n = d d x ( 1 x m ) = − d x x m ( x m ) 2 = − m x m − 1 x 2 m = − m x − m − 1 = n x n − 1. {displayStyle {frac {d}{dx}}x^{n}={frac {d}{dx}}left({frac {1}{x^{m}}}right)={frac {-{frac {d}{dx}}x^{m}}{(x^{m})^{2}}}=-{frac {mx^{m-1}}{x^{2m}}}=-mx^{-m-1}=nx^{n-1}.} From the above results, we can conclude that if are is a rational number, d d x x r = r x r − 1. {displaystyle {frac {d}{dx}}x^{r}=rx^{r-1}.} The power rule is used to find the slope of polynomial functions and any other function that contains an exponent with a real number. In other words, it helps to increase the derivative of a variable to a power (exponent). Fortunately, we have the rule of power to simplify our work enormously, because it allows us to take derivations of functions without having to work beyond the formal definition of limits.
Okay, so what is the rule of power and how to enforce it? or f ′ ( x ) = f ( x ) = e x {displaystyle f`(x)=f(x)=e^{x}} , as required. So, if we apply the string rule to f ( x ) = e r ln x {displaystyle f(x)=e^{rln x}}, we see that Finally, if the function is differentiable at x = 0 {displaystyle x=0}, the limit of definition of the derivation is: Since constants are terms that contain only numbers, These are, in particular, terms without variables. So how can we apply the power rule if there is no variable or exponent we can bring down? If x 0 {displaystyle x<0} 0 {displaystyle -x>0}. This necessarily leads to the same result. Since ( − 1 ) r {displaystyle (-1)^{r}} has no conventional definition, if {displaystyle r} is not a rational number, the irrational power functions for negative bases are not well defined. Since rational powers of -1 with even denominators (in the lowest terms) are not real numbers, these expressions are real only for rational powers with odd denominators (in the lowest terms). To begin with, we should have a working definition of the value of f(x)=x r{displaystyle f(x)=x^{r}}, where {displaystyle r} is a real number. Although it is possible to define value as the limit of a sequence of rational forces approaching irrational power whenever we encounter such power, or as the smallest upper limit of a set of rational forces smaller than the given power, this type of definition is indistinguishable.
It is therefore preferable to use a functional definition, which is usually assumed to be x r = exp ( r ln x ) = e r ln x {displaystyle x^{r}=exp(rln x)=e^{rln x}} for all values of x > 0 {displaystyle x>0}, where exp {displaystyle exp } is the natural exponential function and e {displaystyle e} is the Euler number. [1] [2] First, we can show that the derivative of f ( x ) = e x {displaystyle f(x)=e^{x}} f ′ ( x ) = e x {displaystyle f`(x)=e^{x}}. Simply put, the rule of power lends itself to the following rules of distinction: the general case is really not much more difficult unless we try to do too much. The key is to understand what happens when $ds (x+Delta x)^n$ is multiplied: $$(x+Delta x)^n=x^n + nx^{n-1}Delta x + a_2x^{n-2}Delta x^2+cdots+ +a_{n-1}xDelta x^{n-1} + Delta x^n.$$ We know that multiplication gives a large number of terms in the form $ds x^iDelta x^j$, And indeed that $i + j = n$ in each term. One way to look at this is to understand that one method for multiplying $ds (x+Delta x)^n$ is as follows: In each factor $(x+Delta x)$, select the $x$ or $Delta x$, and then multiply the choices $$n together; Do it in any way you can. For example, for $ds (x+Delta x)^3$, there are eight ways to do this: $$eqalign{ (x+Delta x)(x+Delta x)(x+Delta x)&=xxx + xxDelta x + xDelta x + xDelta x Delta xDelta xcr &qquad+ Delta x xx + Delta xxDelta x + Delta xDelta x + Delta xDelta x + Delta xDelta xcr &= x^3 + x^2Delta x +x^2Delta x +xDelta x^ 2 cr & quad+x^2Delta x +xDelta x^2 +xDelta x^2 +Delta x^3cr &=x^3 + 3x^2Delta x + 3xDelta x^2+Delta x^3cr }$$ Regardless of what $n$ is, there are ways $n $ to select $Delta x$ in one factor and $$x in the remaining factors $n-$1; This means that a term is $ds nx^{n-1}Delta x$.