Calculus Examples

Find dy/dt y=cot(cos(t)^4)^2
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
Differentiate using the chain rule, which states that is where and .
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Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
Differentiate using the Power Rule which states that is where .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Multiply by .
Step 3.4
Differentiate using the chain rule, which states that is where and .
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Step 3.4.1
To apply the Chain Rule, set as .
Step 3.4.2
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Multiply by .
Step 3.6
The derivative of with respect to is .
Step 3.7
Multiply by .
Step 3.8
Simplify.
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Step 3.8.1
Reorder the factors of .
Step 3.8.2
Rewrite in terms of sines and cosines.
Step 3.8.3
Apply the product rule to .
Step 3.8.4
One to any power is one.
Step 3.8.5
Multiply .
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Step 3.8.5.1
Combine and .
Step 3.8.5.2
Combine and .
Step 3.8.6
Rewrite in terms of sines and cosines.
Step 3.8.7
Combine.
Step 3.8.8
Multiply by by adding the exponents.
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Step 3.8.8.1
Multiply by .
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Step 3.8.8.1.1
Raise to the power of .
Step 3.8.8.1.2
Use the power rule to combine exponents.
Step 3.8.8.2
Add and .
Step 3.8.9
Combine and .
Step 3.8.10
Multiply by .
Step 3.8.11
Multiply by .
Step 3.8.12
Separate fractions.
Step 3.8.13
Convert from to .
Step 3.8.14
Multiply by .
Step 3.8.15
Divide by .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .