# Problems of Derivatives

## Exercise 1

Compute the derivative function of $f(x)=x^3-2x^2+1$ at the points $x=-1$, $x=0$ and $x=1$. Explain your result. Find an equation of the tangent line to the graph of $f$ at each of the three given points.

$f’(-1)=7$, $f’(0)=0$ y $f’(1)=-1$.

Tangent line at $x=-1$: $y=-2+7(x+1)$.

Tangent line at $x=0$: $y=1$.

Tangent line at $x=1$: $y=-(x-1)$.

Tangent line at $x=-1$: $y=-2+7(x+1)$.

Tangent line at $x=0$: $y=1$.

Tangent line at $x=1$: $y=-(x-1)$.

## Exercise 2

The pH measures the concentration of hydrogen ions H$^+$ in an aqueous solution. It is defined by $$ \mbox{pH} = -\log_{10}(\mbox{H}^+). $$ Compute the derivative of the pH as a function of the concentration of H$^+$. Study the growth of the pH function.

The pH decreases as the concentration of hydrogen ions H$^+$ increase.