Calculus Examples

Find dy/dx y=(5cos(x))/(1+sin(x))
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the Quotient Rule which states that is where and .
Step 3.3
The derivative of with respect to is .
Step 3.4
Differentiate.
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Step 3.4.1
By the Sum Rule, the derivative of with respect to is .
Step 3.4.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.3
Add and .
Step 3.5
The derivative of with respect to is .
Step 3.6
Raise to the power of .
Step 3.7
Raise to the power of .
Step 3.8
Use the power rule to combine exponents.
Step 3.9
Add and .
Step 3.10
Combine and .
Step 3.11
Simplify.
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Step 3.11.1
Apply the distributive property.
Step 3.11.2
Apply the distributive property.
Step 3.11.3
Simplify the numerator.
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Step 3.11.3.1
Simplify each term.
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Step 3.11.3.1.1
Multiply by .
Step 3.11.3.1.2
Multiply by .
Step 3.11.3.1.3
Rewrite using the commutative property of multiplication.
Step 3.11.3.1.4
Multiply .
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Step 3.11.3.1.4.1
Raise to the power of .
Step 3.11.3.1.4.2
Raise to the power of .
Step 3.11.3.1.4.3
Use the power rule to combine exponents.
Step 3.11.3.1.4.4
Add and .
Step 3.11.3.1.5
Multiply by .
Step 3.11.3.1.6
Multiply by .
Step 3.11.3.2
Factor out of .
Step 3.11.3.3
Factor out of .
Step 3.11.3.4
Factor out of .
Step 3.11.3.5
Apply pythagorean identity.
Step 3.11.3.6
Multiply by .
Step 3.11.4
Reorder terms.
Step 3.11.5
Factor out of .
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Step 3.11.5.1
Factor out of .
Step 3.11.5.2
Factor out of .
Step 3.11.5.3
Factor out of .
Step 3.11.6
Cancel the common factor of and .
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Step 3.11.6.1
Factor out of .
Step 3.11.6.2
Cancel the common factors.
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Step 3.11.6.2.1
Factor out of .
Step 3.11.6.2.2
Cancel the common factor.
Step 3.11.6.2.3
Rewrite the expression.
Step 3.11.7
Move the negative in front of the fraction.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .