Section 3.3 Rules of Differentiation Part 1: Read section 3.3 and answer: 1. 2. 3. Part 2: AWWWWW! Now that we have spent a lot of time using the limit definition of derivative, we will learn the shortcut rules for finding derivatives: Rule 1: Derivative of a Constant Function If f has the constant value f ( x) c , then df d (c ) 0 dx dx f ( x) Example: 4 , what is f ( x) ? Rule 2: General Power Rule If n is a real number, then d n ( x ) nx n 1 dx Note that your book deals with n as a positive integer—in reality, n could be any real no. Examples: Find f '( x) if: 1. f ( x) x 4 2. f ( x) x 3. f ( x) 1 x2 Rule 3: Constant Multiple Rule If u is a differentiable function of x, and c is a constant, then d du (cu ) c dx dx Example: y 5 x 2 , what is dy ? dx Rule 4: Derivative Sum (or Difference) Rule If u and v are differentiable functions of x, then their sum u+v is differentiable at every point where u and v are both differentiable. At such points, d du dv (u v) dx dx dx Find dy 2 if y 5 x3 4 x x dx Derivative of the Natural Exponential Function d x (e ) e x dx Higher Order Derivatives: We can take the derivative of a derivative which would be called the second derivative. The notation for the second derivative would be : Find the second derivative of f ( x) 4 x3 7 x 2 5x 6 Find the third derivative of the above function. f ( x) d 2 y d dy dy y dx 2 dx dx dx We’ve already alluded to the idea that horizontal tangent lines are important. So in order to find where tangent lines are horizontal, we would set the first derivative equal to 0 and solve. Find the equation of the horizontal tangent line(s) to the curve y x3 9 x Other examples: Find the equation of the tangent line to f ( x) x 2 7 x 4 at x=3. What is a “normal line”? Find the equation of the normal line to f ( x) x at x=4.