College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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The Intermediate Value Property of Derivatives

3.2 Differentiation Rules

Ex:- = → =

Ex:- = → =

Ex:- = → = ( ) = ( ) = ( ) =

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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Ex:- ( ) = + → = 5 + 2Products and Quotients

Ex:- For = ( + )( + + ) find a) by applying the product rule, and b) by multiplying the factors to produce a sum of simpler terms to differentiate.Solution:-

Ex:-find for:- 1) = +− 2) = ( − )( + + )Solution:-1)

2)

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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Negative Integer Powers of x

Ex:-

Second- and Higher-Order Derivatives

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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Ex:- = ( − ) + + find and .Solution:-

= ( − ) + + = + + − − − = − = −= − (− ) =

= (− ) = −

Ex:- Find the derivative of all orders of the function =Solution:-

3.3 The Derivative as a Rate of ChangeInstantaneous Rates of Change

Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk

figure below) is

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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Ex:- For ( ) = − + of a body moving on a coordinate line, with s in meters and t in seconds.

a. Find the body’s displacement and average velocity for ≤ ≤ .b. Find the body’s speed and acceleration at the endpoints of the interval.c. When, if ever, during the interval does the body change direction?

Solution:-

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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Ex:-

Solution:-

3.4 Derivatives of Trigonometric Functions

Ex:- find the derivative of ( ) =

Ex:-

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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3.5 The Chain Rule and Parametric Equations

Ex:- in

1. 2.

Solutions:-

1.

2.

“Outside-Inside” Rule

Ex

Ex

Ex

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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Parametric Equations

Slopes of Parametrized Curves

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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Ex:- For the given parametric equations and parameter intervals for the motion of a particle in

the xy-plane, identify the particle’s path by finding a Cartesian equation for it and then graph

the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction

of motion.

1.

2.

3.

Solution:-

1. 2.

3.

Ex:- Find a parametrization for line segment with endpoints (-1,3) and (3,-2).

Solution:-

Ex:- For

Solution:-

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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3.6 Implicit Differentiation

Implicit Differentiation

Rational Powers of Differentiable Functions

Ex:-

Ex:- for:- 1) 2)Solution:-1.

2.

Ex:- for .

Solution:-

Ex:- For verify that the point (-1,3) is on the curve and find the lines that are:-(a) tangent and (b) normal to the curve at the given point.

Solution:-

3.7 Related Rates

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim

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Solution:-

Ex:-

Solution:-

Ex:-

Solution:-

Ex:-