What is the derivative of #x^4#?

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Answer 1 · 2015-09-21T00:31:40

#4x^3#

Use the Power Rule #d/dx(x^n)=nx^(n-1)# with #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#.