Power rule

In calculus, the power rule is used to differentiate functions of the form f ( x ) = x r {displaystyle f(x)=x^{r}} , whenever 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 as it relates a power series with a function's derivatives. In calculus, the power rule is used to differentiate functions of the form f ( x ) = x r {displaystyle f(x)=x^{r}} , whenever 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 as it relates a power series with a function's derivatives. If f : R → R {displaystyle f:mathbb {R} ightarrow mathbb {R} } is a function such that f ( x ) = x r {displaystyle f(x)=x^{r}} , and f {displaystyle f} is differentiable at x {displaystyle x} , then, The power rule for integration, which states that for any real number r ≠ − 1 {displaystyle r eq -1} , may be derived by applying the Fundamental Theorem of Calculus to the power rule for differentiation. To start, we should choose a working definition of the value of f ( x ) = x r {displaystyle f(x)=x^{r}} , where r {displaystyle r} is any real number. Although it is feasible to define the value as the limit of a sequence of rational powers that approach the irrational power whenever we encounter such a power, or as the least upper bound of a set of rational powers less than the given power, this type of definition is not amenable to differentiation. It is therefore preferable to use a functional definition, which is usually taken 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 Euler's number. First, we may demonstrate that the derivative of f ( x ) = e x {displaystyle f(x)=e^{x}} is f ′ ( x ) = e x {displaystyle f'(x)=e^{x}} . 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 demonstrated by Euler. Since the latter two functions are equal for all values of x > 0 {displaystyle x>0} , their derivatives are also equal, whenever either derivative exists, so we have, by the chain rule, or f ′ ( x ) = f ( x ) = e x {displaystyle f'(x)=f(x)=e^{x}} , as was required. Therefore, applying the chain rule to f ( x ) = e r ln ⁡ x {displaystyle f(x)=e^{rln x}} , we see that which simplifies to r x r − 1 {displaystyle rx^{r-1}} . When x < 0 {displaystyle x<0} , we may use the same definition with x r = ( ( − 1 ) ( − x ) ) r = ( − 1 ) r ( − x ) r {displaystyle x^{r}=((-1)(-x))^{r}=(-1)^{r}(-x)^{r}} , where we now have − x > 0 {displaystyle -x>0} . This necessarily leads to the same result. Note that because ( − 1 ) r {displaystyle (-1)^{r}} does not have a conventional definition when r {displaystyle r} is not a rational number, irrational power functions are not well defined for negative bases. In addition, as rational powers of -1 with even denominators (in lowest terms) are not real numbers, these expressions are only real valued for rational powers with odd denominators (in lowest terms).