Use limits to compute the derivative. f'(4) if f ( x ) = 9 x 3

Question

Use limits to compute the derivative. f&#039;(4) if   f  (  x  )  =  9      x          3

trazonombresrg · Accepted Answer

Definition of the derivative of a function:
The derivative of f at x is given by

f^{'} (x) = lim_{△ x \to 0} \frac{f (x + △ x) - f (x)}{△ x}

f (x) = 9 x^{3}

f^{'} (x) = lim_{△ x \to 0} \frac{9 (x + △ x)^{3} - 9 x^{3}}{△ x} = lim_{△ x \to 0} \frac{9 (x^{3} + 3 x^{2} △ x + 3 x (△ x)^{2} + (△ x)^{3}) - 9 x^{3}}{△ x}

= lim_{△ x \to 0} \frac{9 x^{3} + 27 x^{2} △ x + 27 x (△ x)^{2} + 9 (△ x)^{3} - 9 x^{3}}{△ x} = lim_{△ x \to 0} \frac{27 x^{2} △ x + 27 x (△ x)^{2} + 9 (△ x)^{3}}{△ x}

= lim_{△ x \to 0} \frac{9 △ x [3 x^{2} + 3 x (△ x) + (△ x)^{2}]}{△ x} = lim_{△ x \to 0} 9 * (3 x^{2} + 3 x △ x + (△ x)^{2})

= 9 * 3 x^{2} = 27 x^{2}

So,

f^{'} (4) = 27 (4)^{2} = 432

Gorlandint · Accepted Answer

f    ′    (  4  )  =      lim          h      →      0                  f      (      4      +      h      )      −      f      (      4      )        h    =      lim          h      →      0                  (      4      +      h              )                  3                    −              4                  3                      h    =      lim          h      →      0        9            (      4      +      h      −      4      )      (      (      4      +      h              )                  2                    +      4      (      4      +      h      )      +              4                  2                    )        h    =      lim          h      →      0                  9      h      [      (      4      +      h              )                  2                    +      4      (      4      +      h      )      +      16      ]        h    =      lim          h      →      0        9  [  (  4  +  h      )          2        +  4  (  4  +  h  )  +  16  ]  =  9  [  (  4  +  0      )          2        +  4  (  4  +  0  )  +  16  ]  =  432

Use limits to compute the derivative. f'(4) if f(x)=9x^3

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