Derivatives

Derivatives Exercise 1 Using the definition, prove that if $f(x)=1/x$, then $f’(a)=-1/a^{2}$ for $a\neq 0$. Show that the tangent line to $f$ at $(a, 1/a)$ intersects $f$ only at $(a, 1/a)$. Exercise 2 Using the definition, prove that if $f(x)=1/x^{2}$, then $f’(a)=-2/a^{3}$ for $a\neq 0$. Show that the tangent line at $(a, 1/a^{2})$ intersects $f$ at exactly one other point, on the opposite side of the y-axis. Exercise 3 Prove that if $f(x)=\sqrt{x}$, then $f’(a)=1/(2\sqrt{a})$ for $a>0$. (Hint: Rationalize the difference quotient.) ...