Question

hw 12 - higher order derivatives section 2.7: problem 4 (1 point) calculate the 4th derivative of y. y = 3x^8 - 6x^6 + 3x^4 y = y = y = y^(4) =

Explanation:

Step1: Find the first - derivative

Using the power rule $(x^n)'=nx^{n - 1}$, for $y = 3x^{8}-6x^{6}+3x^{4}$, we have $y'=3\times8x^{7}-6\times6x^{5}+3\times4x^{3}=24x^{7}-36x^{5}+12x^{3}$.

Step2: Find the second - derivative

Differentiate $y'$: $y''=24\times7x^{6}-36\times5x^{4}+12\times3x^{2}=168x^{6}-180x^{4}+36x^{2}$.

Step3: Find the third - derivative

Differentiate $y''$: $y'''=168\times6x^{5}-180\times4x^{3}+36\times2x = 1008x^{5}-720x^{3}+72x$.

Step4: Find the fourth - derivative

Differentiate $y'''$: $y^{(4)}=1008\times5x^{4}-720\times3x^{2}+72=5040x^{4}-2160x^{2}+72$.

Answer:

$y' = 24x^{7}-36x^{5}+12x^{3}$, $y'' = 168x^{6}-180x^{4}+36x^{2}$, $y''' = 1008x^{5}-720x^{3}+72x$, $y^{(4)}=5040x^{4}-2160x^{2}+72$