Calculus Examples

Find the Derivative - d/dx y=cos((1-e^(5x))/(1+e^(5x)))
Step 1
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 1.1
To apply the Chain Rule, set as .
Step 1.2
The derivative of with respect to is .
Step 1.3
Replace all occurrences of with .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Differentiate.
Tap for more steps...
Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Add and .
Step 3.4
Since is constant with respect to , the derivative of with respect to is .
Step 4
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 4.1
To apply the Chain Rule, set as .
Step 4.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate.
Tap for more steps...
Step 5.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.2
Multiply by .
Step 5.3
Differentiate using the Power Rule which states that is where .
Step 5.4
Multiply by .
Step 5.5
By the Sum Rule, the derivative of with respect to is .
Step 5.6
Since is constant with respect to , the derivative of with respect to is .
Step 5.7
Add and .
Step 6
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 6.1
To apply the Chain Rule, set as .
Step 6.2
Differentiate using the Exponential Rule which states that is where =.
Step 6.3
Replace all occurrences of with .
Step 7
Differentiate.
Tap for more steps...
Step 7.1
Since is constant with respect to , the derivative of with respect to is .
Step 7.2
Multiply by .
Step 7.3
Differentiate using the Power Rule which states that is where .
Step 7.4
Combine fractions.
Tap for more steps...
Step 7.4.1
Multiply by .
Step 7.4.2
Combine and .
Step 8
Simplify.
Tap for more steps...
Step 8.1
Apply the distributive property.
Step 8.2
Apply the distributive property.
Step 8.3
Apply the distributive property.
Step 8.4
Simplify the numerator.
Tap for more steps...
Step 8.4.1
Simplify each term.
Tap for more steps...
Step 8.4.1.1
Multiply by .
Step 8.4.1.2
Rewrite using the commutative property of multiplication.
Step 8.4.1.3
Multiply by by adding the exponents.
Tap for more steps...
Step 8.4.1.3.1
Move .
Step 8.4.1.3.2
Use the power rule to combine exponents.
Step 8.4.1.3.3
Add and .
Step 8.4.1.4
Multiply by .
Step 8.4.1.5
Multiply by by adding the exponents.
Tap for more steps...
Step 8.4.1.5.1
Move .
Step 8.4.1.5.2
Use the power rule to combine exponents.
Step 8.4.1.5.3
Add and .
Step 8.4.1.6
Multiply by .
Step 8.4.2
Combine the opposite terms in .
Tap for more steps...
Step 8.4.2.1
Add and .
Step 8.4.2.2
Add and .
Step 8.4.3
Subtract from .
Step 8.5
Combine terms.
Tap for more steps...
Step 8.5.1
Move the negative in front of the fraction.
Step 8.5.2
Multiply by .
Step 8.5.3
Multiply by .