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How to find the derivative of #x^4#?

Answer 1

#(d color(red)(x^4))/(dx) = color(blue)(4x^3)#

In general #color(white)("XXX")(d color(red)(ax^b))/(dx) = color(blue)(b*ax^(b-1))#

This is a standard formula, but you could derive it from basic definitions of derivative: #color(white)("XXX")(df(x))/dx = lim_(hrarr0)(f(x+h)-f(x))/h#

In this case: #color(white)("XXX")(dx^4)/(dx) = lim_(hrarr0) ((x+h)^4-x^4)/h#

#color(white)("XXXXX")=lim_(hrarr0)(cancel(x^4)+4x^3h+6x^2h^2+4xh^3+h^4 cancel(- x^4))/h#

#color(white)("XXXXX")=lim_(hrarr0)4x^3+6x^2h+4xh^2+h^3#

#color(white)("XXXXX")=4x^3#

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Answer 2

Emilia Aguilar

To find the derivative of (x^4), we apply the power rule of differentiation.

[ \frac{d}{dx}(x^n) = nx^{n-1} ]

For (x^4), where (n = 4), the derivative is:

[ \frac{d}{dx}(x^4) = 4x^{4-1} = 4x^3 ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.