Find the first and second derivatives of the given function. f(x)=5x3−6x2+6

Question

Find the first and second derivatives of the given function.  f(x)=5x3−6x2+6

Kayla Kline · Accepted Answer

Step 1The given function is&amp;nbsp;f(x)=5x3−6x2+6.The derivative of&amp;nbsp;xn&amp;nbsp;is given by:&amp;nbsp;d&amp;nbsp;dx&amp;nbsp;(xn)=nxn−1therefore the constant&#039;s derivative is zero, that is&amp;nbsp;d&amp;nbsp;dx&amp;nbsp;(c)=0.Step 2To determine the given function&#039;s first derivative&amp;nbsp;f(x)=5x3−6x2+6, differentiate this function with respect to x.f(x)=5x3−6x2+6f′(x)=d&amp;nbsp;dx&amp;nbsp;(6x3−6x2+6)&amp;nbsp;(since,&amp;nbsp;d&amp;nbsp;dx&amp;nbsp;f(x)=f′(x))f′(x)=d&amp;nbsp;dx&amp;nbsp;(5x3)+d&amp;nbsp;dx&amp;nbsp;(−6x2)+d&amp;nbsp;dx&amp;nbsp;(6)f′(x)=5d&amp;nbsp;dx&amp;nbsp;(x3)−6d&amp;nbsp;dx&amp;nbsp;(x2)+d&amp;nbsp;dx&amp;nbsp;(6)f′(x)=5(3x(3−1))−6(2x(2−1))+0f′(x)=15x2−12xThus, first derivative of the given function is&amp;nbsp;f′(x)=15x2−12x.Step 3By contrasting the first derivative, the second derivative of the given function can be found&amp;nbsp;f′(x)=15x2−12x&amp;nbsp;with respect to x.f&amp;nbsp;″(x)=d&amp;nbsp;dx&amp;nbsp;(15x2−12x)&amp;nbsp;(since,&amp;nbsp;d2&amp;nbsp;dx&amp;nbsp;2f(x)=f&amp;nbsp;″(x))&amp;nbsp;

chumants6g · Accepted Answer

d2dx2(5x3−6x2+6)  ddx(5x3−6x2+6)=15x2−12x  =ddx(15x2−12x)  ddx(15x2−12x)=30x−12  =30x-12

Find the first and second derivatives of the given function. f(x)=5x^{3}-6x^{2}+6

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