Page:Calculus Made Easy.pdf/71

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(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.