The derivative of
f(x)=x^{2} was found to be f'(x)=2x.
Here, the derivatives of higher powers of x shall
be investigate to demonstrate a pattern. Note in the algebra shown below, Pascal's triangle is used to expand powers of
(x+h)^{n}.

The function f(x)=x^{3} is an antisymmetic function since f(x)=-f(-x), one can substitute
x with some values to demonstrate this e.g. f(2)=8 and f(-2)=-8, therefore f(2)=-f(-2). A secant line passes
through the points A(x,x^{3}) and B(x+h,(x+h)^{3}). f'(x) is found by taking the limit h → 0.

of the analysis.

The function f(x)=x^{4} is a symmetic function since f(x)=f(-x), one can substitute
x with some values to demonstrate this e.g. f(2)=16 and f(-2)=16, therefore f(2)=f(-2). A secant line passes
through the points A(x,x^{4}) and B(x+h,(x+h)^{4}). f'(x) is found by taking the limit h → 0.

of the analysis.

The function f(x)=x^{5} is an antisymmetic function since f(x)=-f(-x), one can substitute
x with some values to demonstrate this e.g. f(2)=32 and f(-2)=-32, therefore f(2)=-f(-2). A secant line passes
through the points A(x,x^{5}) and B(x+h,(x+h)^{5}). f'(x) is found by taking the limit h → 0.

of the analysis.

The results for f(x)=x^{n} for n=2,3,4 and 5 suggests that the derivative f'(x)=nx^{n-1}, let see
if the pattern holds for a negative value of n.

The function f(x)=x^{-1} is an antisymmetic function since f(x)=-f(-x), one can substitute
x with some values to demonstrate this e.g. f(2)=1/2 and f(-2)=-1/2, therefore f(2)=-f(-2). A secant line passes
through the points A(x,1/x) and B(x+h,1/(x+h)). f'(x) is found by taking the limit h → 0.

of the analysis.

The table summarizes our findings for the derivative of f(x)=x^{n} for several integer n values. The results suggests that the
derivative of f(x)=x^{n} is f'(x)=nx^{n-1} for integer values of n.
The Binomial expansion can be used to prove that the result holds for all positive integer values of n.
However, the derivative rule is valid for all real values of n, including negative, fractional, and irrational values; the proof is beyond
the scope of this page.

f(x) | f'(x) |
---|---|

x^{2} |
2x^{1} |

x^{3} |
3x^{2} |

x^{4} |
4x^{3} |

x^{5} |
5x^{4} |

x^{-1} |
-x^{-2} |

x^{n} |
nx^{n-1} |