3.3 Rules for Differentiation What you’ll learn about Positive integer powers, multiples, sums, and differences Products and Quotients Negative Integer Powers of x Second and Higher Order Derivatives Why? These rules help us find derivatives of functions analytically more efficiently.

Time Saving Rules for Differentiation! Check the Calculus Phobe Power Rule! Rule 1 Derivative of a Constant Function If f is the function with the constant value c, then the derivative of f is zero. Find f’ if f(x) = 4 Rule 2 Power Rule for Positive Integer Powers of x If n is a positive integer, then Find f’ if f(x) = x 3 Find f’ if f(x) = x -3

Power Rule Practice Find dy/dx of the following functions. 1)X 4 2)4X 3 – 2X 2 + 6X – 8 3)X-4 4)4X -3 – 2X X – 8 5)

Homework Page 123 Quick Review 1-10 & Exercises 1-9 odds

Warm Up 1) Find if 2) Find the equation of the tangent line at x = -2 of f(x) = x 3 + 2x 2 – 4. Write it in both point-slope and slope intercept form.

More Rules…. Rule 3 The Constant Multiple Rule If u is a differentiable function of x and c is a constant, then Find f’ if f(x) = 3x 2 Rule 4 The Sum and Difference Rule If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points Find f’ if f(x)=3x 2 – 2x 3

Example 3 Using Calculus and Calculator Graph Using a standard viewing window you should see 3 horizontal points to the curve. Where do these occur? Graph f(x). Horizontal tangents occur at max or minimum points. Use calc, min and calc, max Graph f’(x) Horizontal tangents to f(x) occur when the derivative is zero. Use calc, zero to find the x coordinates.

Example 2 Finding Horizontal Tangents Does the curve of y = x 4 - 2x have any horizontal tangents? If so, where? Horizontal tangents happen when the slope is zero. Find f’, set it equal to zero. Find the horizontal tangents to the curve of y = 3x 2 + 4x - 8

Calculus Phobe Tutorials Rule 5 The Product Rule f ’(uv) = uv’ + vu’ Think: u * v’ + v * u’ Rule 6 The Quotient Rule Think: low de high – high de low all over the square of what’s below!

Example 4: The Product Rule Find f’(x) if f(x) = (x 2 +1)(x 3 + 3)

Example 5 The Quotient Rule Differentiate Support Graphically. Graph your answer and nDeriv (f(x),x,x) The graphs should overlap.

Example 6 Working with Numerical Values Let y = uv be the product of the functions u and v. Find y’(2) if u(2) = 4, u’(2) = -2, v(2) = 2, and v’(2) = 3 Find y’(2) if y = u/v.

Example 7 Find an equation for the line tangent to the curve of at the point (1,2). Support your answer graphically. Would you rather use the power rule or the quotient rule to find y’? Simplify fraction to use power rule Find derivative function Evaluate y’(1) to find slope Write tangent line equation

Example 8 Finding Higher Order Derivatives Find the first 4 derivatives of y’ = y’’ = y’’’ = y (4) =

Homework Page 124 Exercises (odds, skip 19) & 40