Quiz 4 - The Definition of the Derivative

(a) What is the derivative of f(x) at the point c? That is, give the

definition of the derivative.

$$\lim_{h \to 0} \frac{f(c+h)-f(c)}{h}$$

OR

$$\lim_{x \to c} \frac{f(x)-f(c)}{x-c}$$

(b) Use the definition of the derivative to find f'(x) for the following functions:

(1) f(x)=x+7

$$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

$$=\lim_{h \to 0} \frac{(x+h)+7 -(x+7)}{h}= \lim_{h \to 0} \frac{h}{h} = 1$$

(2) $f(x)=\frac{3}{x}$

$$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

$$=\lim_{h \to 0} \frac{\frac{3}{x+h}-\frac{3}{x}}{h} =\lim_{h \to 0} \frac{\frac{3x}{(x)(x+h)}-\frac{3(x+h)}{(x)(x+h)}}{h}$$

$$=\lim_{h \to 0} \frac{3x-3x-3h}{(x)(x+h)h}= \lim_{h \to 0} \frac{-3h}{(x)(x+h)h}$$

$$=\lim_{h \to 0} \frac{-3}{x(x+h)} = \frac{-3}{x^2}$$

(3) $f(x)= \cos x$

$$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

$$=\lim_{h \to 0} \frac{\cos (x+h) - \cos x}{h}$$

$$=\lim_{h \to 0} \frac{ [\cos x \cos h - \sin x \sin h]-\cos x}{h}$$

$$= \lim_{h \to 0} \frac{\cos x \cos h -\cos x -\sin x \sin h }{h}$$

$$=\lim_{h \to 0} \frac{\cos x (\cos h - 1)}{h} + \lim_{h \to 0} \frac{-\sin x \sin h }{h}$$

$$= \cos x (0) -\sin x (1) = - \sin x$$