Common Derivatives and Integrals

Common Derivatives and Integrals

You can navigate to specific sections of this handout by clicking the links below.

Derivative Rules: pg. 1 Integral Formulas: pg. 3 Derivatives Rules for Trigonometric Functions: pg. 4 Integrals of Trigonometric Functions: pg. 5 Special Differentiation Rules: pg. 6 Special Integration Formulas: pg. 7

Derivative Rules:

1. Constant Multiple Rule d [cu] = cu , where c is a constant.

dx

2. Sum and Difference Rule d [u ? v] = u ? v

dx

3. Product Rule d [uv] = uv + vu

dx

4. Quotient Rule

d dx

u v

=

vu - uv v2

5. Constant Rule, d [c] = 0

dx

[ ] 6. Power Rule d u n = nu n-1u dx

7. Power Rule d [x] = 1

dx

8. Derivative Involving Absolute Value d [u ] = u (u),u 0

dx

u

9. Derivative of the Natural Logarithmic Function d [ln u] = u

dx

u

Provided by the Academic Center for Excellence 1

Common Derivatives and Integrals Reviewed June 2008

[ ] 10. Derivative of Natural Exponential Function d eu = euu dx

( )( ) Example 1: Find the derivative of f (x) = 4x - 3x2 3 + 2x2

Since there are two polynomials multiplied by each other, apply the third derivative rule, the Product Rule, to the problem.

This is the result of the Product Rule:

[ ] [ ] ( ) ( ) f (x) = 4x - 3x2 d 3 + 2x2 + 3 + 2x2 d 4x - 3x2

dx

dx

Now, take the derivative of each term inside of the brackets. Multiple derivative rules are used, including the Sum and Difference Rule, Constant Rule, Constant Multiple Rule, and Power Rule. When applied, the result is:

f (x) = (4x - 3x2 )(0 + 4x) + (3 + 2x2 )(4 - 6x)

Simplify:

f (x) = (4x - 3x2 )(4x) + (3 + 2x2 )(4 - 6x)

Multiply the polynomials by each other:

( ) ( ) f (x) = 16x2 -12x3 + 12 + 8x2 -18x -12x3

f (x) = 16x2 -12x3 + 12 + 8x2 -18x -12x3

Combine like terms to get a simplified answer:

f (x) = -24x3 + 24x2 -18x + 12

Provided by the Academic Center for Excellence 2

Common Derivatives and Integrals Reviewed June 2008

Integral Formulas: Indefinite integrals have +C as an arbitrary constant.

1. kf (u)du = k f (u)du , where k is a constant. 2. [ f (u) ? g(u)]du = f (u)du ? g(u)du

3. du = u + C

4. u ndu = u n+1 + C, n -1

n +1

5.

du u

= ln u

+C

6. eu du = eu + C

( ) Example 2: Evaluate 4x2 - 5x3 + 12 dx

To evaluate this problem, use the first four Integral Formulas. First, use formula 2 to make the large integral into three smaller integrals:

( ) 4x2 - 5x3 + 12 dx = 4x2dx - 5x3dx + 12dx

Second, pull out the constants by using formula 1:

= 4 x2dx - 5 x3dx + 12 dx

Now find each integral using formulas 3 and 4:

=

4

x 2

2 +1

+1

-

5

x 3

3+1

+1

+

12(x

)

+

C

Provided by the Academic Center for Excellence

3

Common Derivatives and Integrals

Although three integrals have been removed, only one constant C is needed because C represents all unknown constants. Therefore multiple C's can be combined into just one C. To get the final answer, simplify the expression:

=

4

x3 3

-

5

x4 4

+

12x

+

C

= 4 x3 - 5 x4 + 12x + C 34

Derivatives Rules for Trigonometric Functions:

1. d [sin(u)] = (cos(u))u

dx

2. d [cos(u)] = -(sin(u))u

dx

3. d [tan(u)] = (sec2 (u))u dx

4. d [cot(u)] = -(csc2 (u))u dx

5. d [sec(u)] = (sec(u)tan(u))u

dx

6. d [csc(u)] = -(csc(u)cot(u))u

dx

Example 3: Find the derivative of

f (x) =

sin(x) cos(x)

When finding the derivatives of trigonometric functions, non-trigonometric derivative rules are often incorporated, as well as trigonometric derivative rules. Looking at this function, one can see that the function is a quotient. Therefore, use derivative rule 4 on page 1, the Quotient Rule, to start this problem:

Provided by the Academic Center for Excellence

4

Common Derivatives and Integrals

(cos(x)) d [sin(x)]- (sin(x)) d [cos(x)]

f (x) =

dx

dx

cos2 (x)

Now use trigonometric derivative rules 1 and 2 to get:

f

(x)

=

(cos(x))(cos(x)) - (sin(x))(- cos2 (x)

sin(x))

Once the multiplication has been completed in the numerator of the fraction, the result is:

f

(x)

=

(sin

2

(x) +

cos 2

cos 2

(x)

(x))

Remember that sin 2 (x) + cos2 (x) = 1; therefore, substitute 1 for sin2 (x) + cos2 (x) in the

answer. The final result is:

f

(x)

=

1 cos 2

(x)

f (x) = sec2 (x)

Integrals of Trigonometric Functions:

1. sin(u)du = - cos(u) + C

2. cos(u)du = sin(u) + C

3 tan(u)du = - ln cos(u) + C

4. cot(u)du = ln sin(u) + C

5. sec(u)du = ln sec(u) + tan(u) + C

Provided by the Academic Center for Excellence

5

Common Derivatives and Integrals

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download