Derivatives, Properties, and Formulas

The derivative of a function $f$ is a new function $f '(x)$ which gives the slope of the line tangent to the graph of $f$ at each point $x$. To find the slope of the tangent line at a particular point $( c, f(c) )$ on the graph of $f$, we should first calculate the derivative $f '(x)$ and then evaluate the function $f '(x)$ at the point $x = c$ to get the number $f '(c)$. If you mistakenly evaluate $f$ first, you get a number $f(c)$, and the derivative of a constant is always equal to 0. 

Example 10: Determine the slope of the line tangent to $f(x) = 3x + sin(x)$ at $(0, f(0) )$ and $(1, f(1 ))$: 

Solution:$f '(x) = D( 3x + sin(x) ) = D(3x) + D( sin(x) ) = 3 + cos(x)$ When $x = 0$, the graph of $y = 3x + sin(x)$ goes through the point $( 0, 3(0)+sin(0) )$ with slope $f '(0) = 3 + cos(0) = 4$. When $x = 1$, the graph goes through the point $( 1, 3(1)+sin(1) ) = (1, 3.84)$ with slope $f '(1) = 3 + cos(1) ≈ 3.54$. 

Practice 11: Where do $f(x) = x^2 – 10x + 3$ and $g(x) = x^3 – 12x$ have a horizontal tangent lines ?