What is the derivative of $x^{4}$ $x^4$ ?

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Answer 1 · 2015-09-21T00:31:40

$4 x^{3}$ $4x^3$

Use the Power Rule $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ $d/dx(x^n)=nx^(n-1)$ with $n = 4$ $n=4$ .

From first principles: from the definition of the derivative, the Binomial Theorem, and some algebra, you can also write:

$d/dx(x^4)=lim_{h->0}((x+h)^4-x^4)/h$

$=lim_{h->0}(x^4+4x^3h+6x^2h^2+4xh^3+h^4-x^4)/h$

$=lim_{h->0}(h(4x^3+6x^2h+4xh^2+h^3))/h$

$=lim_{h->0}(4x^3+6x^2h+4xh^2+h^3)$ .

For any fixed $x$ , this is a continuous function of $h$ and the limit can now be evaluated by substitution:

$d/dx(x^4)=4x^3+6x^2*0+4x*0^2+0^3=4x^3$ .

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