Page:Calculus Made Easy.pdf/71

 Examples. Now let us try $$y = f(x) = 7x^4 + 3.5x^3 - \frac{1}{2}x^2 + x - 2$$.
 * $$\frac{dy}{dx}= f'(x) = 28x^3 + 10.5x^2 - x + 1$$,
 * $$\frac{d^2y}{dx^2}= f''(x) = 84x^2 + 21x - 1$$,
 * $$\frac{d^3y}{dx^3}= f'''(x) = 168x + 21$$,
 * $$\frac{d^4y}{dx^4}= f(x) = 168$$,
 * $$\frac{d^5y}{dx^5}= f'(x) = 0$$.

In a similar manner if $$y = \phi(x) = 3x(x^2 - 4)$$,
 * $$\phi'(x)=\frac{dy}{dx} = 3\left[x \times 2x + (x^2 - 4) \times 1\right] =3(3x^2-4)$$,
 * $$\phi''(x)=\frac{d^2y}{dx^2}=3 \times 6x= 18x$$,
 * $$\phi'''(x)=\frac{d^3y}{dx^3}=18$$,
 * $$\phi(x)=\frac{d^4y}{dx^4}=0$$.

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