Remembering the limit laws, we recall the following theorem.

**Theorem 1**. Suppose that functions f and g are defined on some open interval (a, b) and f ‘(x) and g ‘(x) exist at each point x in (a, b). Then

To emphasize the fact that the derivatives are taken with respect to the independent variable x, we use the following notation, as is customary:

Based on previous Theorem 1. and the definition of the derivative, we get the following theorem.

**Theorem 2.** Suppose that functions f and g are defined on some open interval (a, b) and f ‘(x) and g ‘(x) exist at each point x in (a, b). Then

**Exercise 1**. Compute the following derivative:

* *

*Solution*. Using the sum, difference and constant multiple rules, we get:

**Exercise 2.** Compute the following derivative:

*Solution*. Using the sum, difference and constant multiple rules, we get:

**Exercise 3**. Compute the following derivative:

*Solution*. Using the sum and product rules, we get:

**Exercise 4**. Compute the following derivative:

*Solution*. Using the sum and quotient rules, we get:

**Exercise 5.** Suppose Use the definition of derivative to find f’(x).

*Solution*. Using the definition of derivative, we get:

**Exercise 6.** Suppose Use the definition of derivative to find f’(x).

*Solution*. Using the definition of derivative, we get:

Suppose we have two functions, u and y, related by the equations: u = g (x) and y = f (u). Then y = (f _{°} g)(x) = f (g (x)).

The chain rule deals with the derivative of the composition and may be stated as the following theorem:

**Theorem 3**. (The Chain Rule). Suppose that g is defined in an open interval I containing c, and f is defined in an open interval J containing g(c), such that g(x) is in J for all x in I. If g is differentiable at c, and f is differentiable at g(c), then the composition (f _{°} g) is differentiable at c and

**Exercise 7**. Let Find (f _ g)’(x).

*Solution*. Using the chain rule to derive, we get:

**Exercise 8**. Let Find f’(x).

*Solution*. Using the chain rule, we get:

**Exercise 9**. Let Find f’(x).

*Solution*. Using the chain rule, we get:

**Exercise 10**. Let Find f’(x).

*Solution*. Using the chain rule, we get:

Recall that the line tangent to the graph of a function f at (c, f(c)) has slope f ‘(c) and has equation:

**Exercise 11**. Find the equation of the tangent line for the graph of the following function f at the given point c:

*Solution*. We calculate the function f at the given point c=1:

Then using the derivative rules, we get:

**Exercise 12**. Find the equation of the tangent line for the graph of the following function f at the given point c:

*Solution*. We calculate the function f at the given point c=0:

Then using the derivative rules, we get:

*1*. Functions and their graph. One–one and onto functions. Composite and inverse functions.

*2*. Linear inequalities. Systems of linear inequalities

*3*. Polynomial inequalities. Rational inequalities. Absolute-value inequalities

*6*. Differentiation. Derivative rules, the chain rule.

*7*. Application of differentiation: L'Hospital's Rule

*8*. Vertical, Horizontal and Slant asymptotes

*9*. Higher Order Derivatives. Applications of differentiation: local and absolute extremes of a function

*10*. Curve Sketching

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