Scribd
Upload
14 views3 pages

Derivatives

The document provides a comprehensive list of derivative and integration formulas, including rules such as the Power Rule, Sum Rule, Product Rule, and Quotient Rule for derivatives, as well as corresponding integration rules. It details derivatives of various functions like sine, cosine, exponential, and logarithmic functions, along with higher-order derivatives and implicit differentiation. Additionally, it covers integration techniques including the Power Rule for integration, integration of exponential and trigonometric functions, and methods like substitution and integration by parts.

Uploaded by

Roshan R.S
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
14 views3 pages

Derivatives

The document provides a comprehensive list of derivative and integration formulas, including rules such as the Power Rule, Sum Rule, Product Rule, and Quotient Rule for derivatives, as well as corresponding integration rules. It details derivatives of various functions like sine, cosine, exponential, and logarithmic functions, along with higher-order derivatives and implicit differentiation. Additionally, it covers integration techniques including the Power Rule for integration, integration of exponential and trigonometric functions, and methods like substitution and integration by parts.

Uploaded by

Roshan R.S
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 3

Derivative and Integration Formulas

Derivative Formulas
1. Power Rule:
d n
x = nxn−1
dx
2. Sum Rule:
d d d
(f (x) + g(x)) = f (x) + g(x)
dx dx dx
3. Product Rule:
d
[f (x)g(x)] = f ′ (x)g(x) + f (x)g ′ (x)
dx
4. Quotient Rule:
f ′ (x)g(x) − f (x)g ′ (x)
 
d f (x)
=
dx g(x) g(x)2
5. Chain Rule:
d
[f (g(x))] = f ′ (g(x)) · g ′ (x)
dx
6. Derivative of Sine:
d
sin(x) = cos(x)
dx
7. Derivative of Cosine:
d
cos(x) = − sin(x)
dx
8. Derivative of Exponential Function:
d x
e = ex
dx
9. Derivative of Natural Logarithm:
d 1
ln(x) =
dx x
10. Derivative of Logarithmic Functions:
d 1
loga (x) =
dx x ln(a)
11. Derivative of tan(x):
d
tan(x) = sec2 (x)
dx

1
12. Derivative of sec(x):
d
sec(x) = sec(x) tan(x)
dx
13. Derivative of cot(x):
d
cot(x) = − csc2 (x)
dx
14. Derivative of csc(x):
d
csc(x) = − csc(x) cot(x)
dx
15. Implicit Differentiation:

dy − ∂F
∂x
= ∂F
dx ∂y

16. Higher-order Derivatives: First derivative:

f ′ (x)

Second derivative:
f ′′ (x)
Third derivative:
f (3) (x)
General nth derivative:
f (n) (x)
17. Derivative of a Composite Function:
d
f (g(x)) = f ′ (g(x)) · g ′ (x)
dx

Integration Formulas
1. Power Rule for Integration:
xn+1
Z
xn dx = + C, n ̸= −1
n+1
2. Sum Rule for Integration:
Z Z Z
(f (x) + g(x)) dx = f (x)dx + g(x)dx

3. Integration of Exponential Function:


Z
ex dx = ex + C

2
4. Integration of Natural Logarithm:
Z
ln(x)dx = x ln(x) − x + C

5. Integration of Sine:
Z
sin(x)dx = − cos(x) + C

6. Integration of Cosine:
Z
cos(x)dx = sin(x) + C

7. Integration of sec2 (x):


Z
sec2 (x)dx = tan(x) + C

8. Integration of csc2 (x):


Z
csc2 (x)dx = − cot(x) + C

9. Integration of sec(x) tan(x):


Z
sec(x) tan(x)dx = sec(x) + C

10. Integration of csc(x) cot(x):


Z
csc(x) cot(x)dx = − csc(x) + C

11. Substitution Rule:


Z
f (g(x))g ′ (x)dx = F (g(x)) + C

12. Integration by Parts:


Z Z
u dv = uv − v du

13. Integration of Rational Functions (Partial Fractions):


Z
P (x)
dx (where the degree of P (x) is less than the degree of Q(x))
Q(x)

You might also like