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Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = x^4 $

$4 x^{3}$

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suppose you want to find the derivative of the function using the definition of derivatives. And here we have F of X, which is equal to X. Rays to the 4th power. Now by definition the derivative of F of X is given by F prime of X. This is equal to The limit as H approaches zero off F of expose each minus F of X over H. Now if F of X is equal to X. Rays to the 4th power, then F of exports H. This would be Expose Age race to the 4th power. And so we have limits. as a jew approaches zero of expose H raised to the fourth power minus X to the fourth power over each. And then from here we want to factor out the numerator and we have Limit as H approaches zero of X plus H squared minus X squared. These times X plus H squared plus X squared and then this all over H. Factoring the first factor we have limit as Asia approaches zero of we have X plus H minus X times X plus H plus X. Times expose H squared plus X squared. This all over each. Now simplifying we have limit as h approaches zero of age. Times We have two x plus H times X plus age squared plus X squared face. All over each. Now in here we can cancel out the age and you're left with the limit as a trough approaches zero of two x plus each times X was H squared plus X squared. Now evaluating at the HR go zero, we have two x plus zero times extra zero squared plus X square. This gives us two X times two x squared which is just four x rays to the third power. And so this is the derivative of the function F of X, which is equal to X to the 4th power.