The Constant and Power Rules for Derivatives Calculus 1 Ms. Penabad 2014

In today’s lesson we will cover: Theorem 3.2: Constant Rule Example 1 Example 2 Example 3 Theorem 3.3: Power Rule

Constant Rule Theorem 3.2: Constant Rule If c is a real number, then .

Let’s evaluate the following derivative: Example 1 Let’s evaluate the following derivative:

Let’s evaluate the following derivative: Example 1 Let’s evaluate the following derivative: RECALL: Constant Rule

Let’s evaluate the following derivative: Example 1 RECALL: Constant Rule Let’s evaluate the following derivative: Since 2014 is a real number, it is considered to be a constant. By the Constant Rule, the derivative of ANY constant will always equal 0.

Let’s evaluate the following derivative: Example 2 Let’s evaluate the following derivative:

Let’s evaluate the following derivative: Example 2 Let’s evaluate the following derivative: RECALL: Constant Rule

Let’s evaluate the following derivative: Example 2 RECALL: Constant Rule Let’s evaluate the following derivative: Step 1: if visualization needed: simplify exponent (-8) by moving e to the denominator.

Let’s evaluate the following derivative: Example 2 RECALL: Constant Rule Let’s evaluate the following derivative: Step 2: since is a constant, by the Constant Rule, . Step 1: if visualization needed: simplify exponent (-8) by moving e to the denominator.

Let’s evaluate the following derivative: Example 3 Let’s evaluate the following derivative:

Let’s evaluate the following derivative: Example 3 Let’s evaluate the following derivative: RECALL: Constant Rule

Example 3 Let’s evaluate the following derivative: RECALL: Constant Rule Let’s evaluate the following derivative: Step 1: if visualization needed: simplify fraction.

Example 3 Let’s evaluate the following derivative: RECALL: Constant Rule Let’s evaluate the following derivative: Step 2: since 2.6 is a constant, by the Constant Rule, the answer is 0. Step 1: if visualization needed: simplify fraction.

Power Rule Theorem 3.2: Power Rule If n is a positive integer, then .

Let’s evaluate the following derivative: Example 1 Let’s evaluate the following derivative:

Let’s evaluate the following derivative: Example 1 Let’s evaluate the following derivative: RECALL: Power Rule

Let’s evaluate the following derivative: Example 1 RECALL: Power Rule Let’s evaluate the following derivative:

Example 1 Let’s evaluate the following derivative: RECALL: Power Rule Step 1: notice x has a coefficient of 1

Example 1 Let’s evaluate the following derivative: RECALL: Power Rule Step 1: notice x has a coefficient of 1 Step 2: move the 9 out front Step 3: subtract 1 from the exponent

Example 1 Let’s evaluate the following derivative: RECALL: Power Rule Step 4: simplify coefficient (1·9)=9 exponent (9-1)=8 Step 1: notice x has a coefficient of 1 Step 2: move the 9 out front Step 3: subtract 1 from the exponent

Let’s evaluate the following derivative: Example 2 Let’s evaluate the following derivative:

Let’s evaluate the following derivative: Example 2 Let’s evaluate the following derivative: RECALL: Power Rule

Let’s evaluate the following derivative: Example 2 RECALL: Power Rule Let’s evaluate the following derivative:

Example 2 Let’s evaluate the following derivative: RECALL: Power Rule Step 1: notice x has an exponent of 1

Example 2 Let’s evaluate the following derivative: RECALL: Power Rule Step 1: notice x has an exponent of 1 Step 2: move the 1 out front Step 3: subtract 1 from the exponent

Example 2 Let’s evaluate the following derivative: RECALL: Power Rule Step 4: simplify (1-1)=0 Step 1: notice x has an exponent of 1 Step 2: move the 1 out front Step 3: subtract 1 from the exponent

Example 2 Let’s evaluate the following derivative: RECALL: Power Rule Step 4: simplify (1-1)=0 Step 5: simplify =1 Step 1: notice x has an exponent of 1 Step 2: move the 1 out front Step 3: subtract 1 from the exponent

Let’s evaluate the following derivative: Example 3 Let’s evaluate the following derivative:

Let’s evaluate the following derivative: Example 3 Let’s evaluate the following derivative: RECALL: Power Rule

Example 3 Let’s evaluate the following derivative: RECALL: Power Rule Step 1: move the 8 out front Step 2: subtract 1 from the exponent

Example 3 Let’s evaluate the following derivative: RECALL: Power Rule Step 3: simplify coefficient (2·8)=16 exponent (8-1)=7 Step 1: move the 8 out front Step 2: subtract 1 from the exponent

Now…LET’S PRACTICE