Calculus Examples

Find the 2nd Derivative sec(x)^2
Step 1
Find the first derivative.
Tap for more steps...
Step 1.1
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 1.1.1
To apply the Chain Rule, set as .
Step 1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3
Replace all occurrences of with .
Step 1.2
The derivative of with respect to is .
Step 1.3
Raise to the power of .
Step 1.4
Raise to the power of .
Step 1.5
Use the power rule to combine exponents.
Step 1.6
Add and .
Step 2
Find the second derivative.
Tap for more steps...
Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Product Rule which states that is where and .
Step 2.3
The derivative of with respect to is .
Step 2.4
Multiply by by adding the exponents.
Tap for more steps...
Step 2.4.1
Use the power rule to combine exponents.
Step 2.4.2
Add and .
Step 2.5
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 2.5.1
To apply the Chain Rule, set as .
Step 2.5.2
Differentiate using the Power Rule which states that is where .
Step 2.5.3
Replace all occurrences of with .
Step 2.6
Move to the left of .
Step 2.7
The derivative of with respect to is .
Step 2.8
Raise to the power of .
Step 2.9
Raise to the power of .
Step 2.10
Use the power rule to combine exponents.
Step 2.11
Add and .
Step 2.12
Raise to the power of .
Step 2.13
Raise to the power of .
Step 2.14
Use the power rule to combine exponents.
Step 2.15
Add and .
Step 2.16
Simplify.
Tap for more steps...
Step 2.16.1
Apply the distributive property.
Step 2.16.2
Multiply by .
Step 2.16.3
Reorder terms.
Step 3
Find the third derivative.
Tap for more steps...
Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
Tap for more steps...
Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Differentiate using the Product Rule which states that is where and .
Step 3.2.3
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 3.2.3.1
To apply the Chain Rule, set as .
Step 3.2.3.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3.3
Replace all occurrences of with .
Step 3.2.4
The derivative of with respect to is .
Step 3.2.5
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 3.2.5.1
To apply the Chain Rule, set as .
Step 3.2.5.2
Differentiate using the Power Rule which states that is where .
Step 3.2.5.3
Replace all occurrences of with .
Step 3.2.6
The derivative of with respect to is .
Step 3.2.7
Multiply by by adding the exponents.
Tap for more steps...
Step 3.2.7.1
Move .
Step 3.2.7.2
Use the power rule to combine exponents.
Step 3.2.7.3
Add and .
Step 3.2.8
Move to the left of .
Step 3.2.9
Raise to the power of .
Step 3.2.10
Raise to the power of .
Step 3.2.11
Use the power rule to combine exponents.
Step 3.2.12
Add and .
Step 3.2.13
Multiply by by adding the exponents.
Tap for more steps...
Step 3.2.13.1
Move .
Step 3.2.13.2
Multiply by .
Tap for more steps...
Step 3.2.13.2.1
Raise to the power of .
Step 3.2.13.2.2
Use the power rule to combine exponents.
Step 3.2.13.3
Add and .
Step 3.2.14
Move to the left of .
Step 3.3
Evaluate .
Tap for more steps...
Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 3.3.2.1
To apply the Chain Rule, set as .
Step 3.3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.3.2.3
Replace all occurrences of with .
Step 3.3.3
The derivative of with respect to is .
Step 3.3.4
Raise to the power of .
Step 3.3.5
Use the power rule to combine exponents.
Step 3.3.6
Add and .
Step 3.3.7
Multiply by .
Step 3.4
Simplify.
Tap for more steps...
Step 3.4.1
Apply the distributive property.
Step 3.4.2
Combine terms.
Tap for more steps...
Step 3.4.2.1
Multiply by .
Step 3.4.2.2
Multiply by .
Step 3.4.2.3
Add and .
Step 4
Find the fourth derivative.
Tap for more steps...
Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Evaluate .
Tap for more steps...
Step 4.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2.2
Differentiate using the Product Rule which states that is where and .
Step 4.2.3
The derivative of with respect to is .
Step 4.2.4
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 4.2.4.1
To apply the Chain Rule, set as .
Step 4.2.4.2
Differentiate using the Power Rule which states that is where .
Step 4.2.4.3
Replace all occurrences of with .
Step 4.2.5
The derivative of with respect to is .
Step 4.2.6
Multiply by by adding the exponents.
Tap for more steps...
Step 4.2.6.1
Use the power rule to combine exponents.
Step 4.2.6.2
Add and .
Step 4.2.7
Raise to the power of .
Step 4.2.8
Use the power rule to combine exponents.
Step 4.2.9
Add and .
Step 4.2.10
Raise to the power of .
Step 4.2.11
Raise to the power of .
Step 4.2.12
Use the power rule to combine exponents.
Step 4.2.13
Add and .
Step 4.3
Evaluate .
Tap for more steps...
Step 4.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.2
Differentiate using the Product Rule which states that is where and .
Step 4.3.3
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 4.3.3.1
To apply the Chain Rule, set as .
Step 4.3.3.2
Differentiate using the Power Rule which states that is where .
Step 4.3.3.3
Replace all occurrences of with .
Step 4.3.4
The derivative of with respect to is .
Step 4.3.5
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 4.3.5.1
To apply the Chain Rule, set as .
Step 4.3.5.2
Differentiate using the Power Rule which states that is where .
Step 4.3.5.3
Replace all occurrences of with .
Step 4.3.6
The derivative of with respect to is .
Step 4.3.7
Raise to the power of .
Step 4.3.8
Raise to the power of .
Step 4.3.9
Use the power rule to combine exponents.
Step 4.3.10
Add and .
Step 4.3.11
Multiply by by adding the exponents.
Tap for more steps...
Step 4.3.11.1
Move .
Step 4.3.11.2
Multiply by .
Tap for more steps...
Step 4.3.11.2.1
Raise to the power of .
Step 4.3.11.2.2
Use the power rule to combine exponents.
Step 4.3.11.3
Add and .
Step 4.3.12
Move to the left of .
Step 4.3.13
Multiply by by adding the exponents.
Tap for more steps...
Step 4.3.13.1
Move .
Step 4.3.13.2
Use the power rule to combine exponents.
Step 4.3.13.3
Add and .
Step 4.3.14
Move to the left of .
Step 4.4
Simplify.
Tap for more steps...
Step 4.4.1
Apply the distributive property.
Step 4.4.2
Apply the distributive property.
Step 4.4.3
Combine terms.
Tap for more steps...
Step 4.4.3.1
Multiply by .
Step 4.4.3.2
Multiply by .
Step 4.4.3.3
Multiply by .
Step 4.4.3.4
Add and .
Step 4.4.4
Reorder terms.