Examples.

Now let us try ${\displaystyle y=f(x)=7x^{4}+3.5x^{3}-{\frac {1}{2}}x^{2}+x-2}$.

${\displaystyle {\frac {dy}{dx}}=f'(x)=28x^{3}+10.5x^{2}-x+1}$,

${\displaystyle {\frac {d^{2}y}{dx^{2}}}=f''(x)=84x^{2}+21x-1}$,

${\displaystyle {\frac {d^{3}y}{dx^{3}}}=f'''(x)=168x+21}$,

${\displaystyle {\frac {d^{4}y}{dx^{4}}}=f''''(x)=168}$,

${\displaystyle {\frac {d^{5}y}{dx^{5}}}=f'''''(x)=0}$.

In a similar manner if ${\displaystyle y=\phi (x)=3x(x^{2}-4)}$,

${\displaystyle \phi '(x)={\frac {dy}{dx}}=3\left[x\times 2x+(x^{2}-4)\times 1\right]=3(3x^{2}-4)}$,

${\displaystyle \phi ''(x)={\frac {d^{2}y}{dx^{2}}}=3\times 6x=18x}$,

${\displaystyle \phi '''(x)={\frac {d^{3}y}{dx^{3}}}=18}$,

${\displaystyle \phi ''''(x)={\frac {d^{4}y}{dx^{4}}}=0}$.

Exercises IV. (See page 255 for Answers.) Find ${\displaystyle {\frac {dy}{dx}}}$ and ${\displaystyle {\frac {d^{2}y}{dx^{2}}}}$ for the following expressions:

(1) ${\displaystyle y=17x+12x^{2}}$.

(2) ${\displaystyle y={\frac {x^{2}+a}{x+a}}}$.

(3) ${\displaystyle y=1+{\frac {x}{1}}+{\frac {x^{2}}{1\times 2}}+{\frac {x^{3}}{1\times 2\times 3}}+{\frac {x^{4}}{1\times 2\times 3\times 4}}}$.

(4) Find the 2nd and 3rd derived functions in the Exercises III. (p. 46), No. 1 to No. 7, and in the Examples given (p. 41), No. 1 to No. 7.