Guided Notes for Hill’s Fall 2005 Math 229  Lecture 9  Page 1

Lecture 9: Sections 10.2, 11.3, and 11.5

Section 10.2 — Combining Functions

Read section 10.2. This material is expected to be review, and we will skip most of it in class. We will focus on the composition of functions.

We need to be able to recognize when one function can be broken down into simpler functions. Definition: Let f and g be functions. Then the composition of g and f is the function g ∘ f defined by

(g ∘ f) (x) = g [f(x)] .

Example: Let f(x) = 2x − 3 and g(x) = x^{2} + 2. Find the following.

(a) (f ∘ g) (x)  (b) (g ∘ f) (x)

Note: Function composition is not commutative. Example: Express each of the following functions as a composition of simpler functions.

(a) f(x) = \sqrt{x^{2} − 3}  (b) f(x) = \frac{3}{\sqrt{x^{5} − 2x + 1}}  (c) f(x) = (2x^{2} − 3x + 5)^{4}

Note: The goal for our purposes is for each of the simpler functions to be ones whose derivative is easily found. Section 11.3 — The Chain Rule

We could take the derivative of y = (2x^{2} − 3x + 5)^{4} by multiplying it out first or by repeated use of the product rule, but either procedure is tedious. There is a better way. Properties: The chain rule: If h(x) = (g ∘ f) (x) = g [f(x)] then

h′(x) = g′ [f(x)] · f′(x).

Note: The derivative can be thought of as g′ (original “inside”) · (derivative of the “inside”).

Guided Notes for Hill’s Fall 2005 Math 229  Lecture 9  Page 2

Example: Find the derivative of each of the following functions.

(a) y = (2x^{2} − 3x + 5)^{4} ; (b) y = \sqrt{x^{2} − 3} ; (c) y = \frac{3}{\sqrt{x^{5} − 2x + 1}}  {(d) y = (3x − 1)^{4} (2x^{2} + 5)^{5} \atop (e) y = (\frac{3x − 1}{2x + 5})^{3}}

Example: #68 in section 11.3. Section 11.5 — Higher-Order Derivatives Definition: The second derivative of a function f is the derivative of its derivative.

Notations: f′′(x) = \frac{d^{2}f}{dx^{2}} = y′′ = D^{2}y.

Note that f′′(x) = \frac{d}{dx} [f′(x)].

Guided Notes for Hill’s Fall 2005 Math 229  Lecture 9  Page 3

Definition: The nth derivative of a function is the derivative of its (n − 1)st derivative.

Notations: f^{(n)}(x) = \frac{d^{n}f}{dx^{n}} = y^{(n)} = D^{n}y.

Example: Find all derivatives of P (x) = 3x^{4} − 11x^{3} + 12x^{2} − 3x + 2.

Example: Find the second derivative of y = \frac{5x + 3}{2x − 1}.

Properties: If s(t) gives the position at time t, then s′(t) = v(t) gives the velocity and s′′(t) = a(t) gives the acceleration at time t. Example: #38 in section 11.5

Continue with Lab 9.

Do the homework for sections 10.2, 11.3, and 11.5.

Read section 11.4 for Thursday.