3. Differentiation Rules

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Presentation transcript:

3. Differentiation Rules

Notation for Derivative is a noun. It means “the derivative of y with respect to x” is a verb. It means “take the derivative with respect to x” of the expression that follows.

1. Constant Rule The derivative of a constant function is 0 (Think about the function y=2 to discover why) Example 1 – find the derivative of the following functions

2. Linear Rule The derivative of a linear function is m. (Think about the function y=3x+1 to discover why) Example 2 – find the derivative of the following functions

3. Power(ful) Rule Bring down the exponent and subtract one from the exponent Example 3 – Find the derivative of the following functions. It is important to rewrite first so x is only in numerator!

Example 4 Find an equation of the tangent line and normal line to the graph f(x)=3x2 when x = -2.

Example 5 Find the coordinates where the function f(x) = -5x2 has tangent lines with the following slopes a. m= -3 b. m= 0 c. m=2/3

Rewriting As we have seen, the ability to rewrite functions is very important when using the power rule!! Example 6 – Rewrite, evaluate, then simplify

Example 7 Find the coordinates and equations of any horizontal tangents to the curve y = x4 - 2x2 + 2

4. Sin and cos rules Example 8 – find the derivatives of the following functions.

Example 9 Find the slope of the graph of f(x) = sin x at the origin and at

Example 10 – Our old friend – the piecewise function If , find values of a and b such that f(x) is differentiable everywhere