Calculus Examples

Find the Tangent Line at (π,0) y=sin(sin(x)) , (pi,0)
,
Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
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Step 1.1
Differentiate using the chain rule, which states that is where and .
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Step 1.1.1
To apply the Chain Rule, set as .
Step 1.1.2
The derivative of with respect to is .
Step 1.1.3
Replace all occurrences of with .
Step 1.2
The derivative of with respect to is .
Step 1.3
Reorder the factors of .
Step 1.4
Evaluate the derivative at .
Step 1.5
Simplify.
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Step 1.5.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 1.5.2
The exact value of is .
Step 1.5.3
Multiply by .
Step 1.5.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.5.5
The exact value of is .
Step 1.5.6
The exact value of is .
Step 1.5.7
Multiply by .
Step 2
Plug the slope and point values into the point-slope formula and solve for .
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Step 2.1
Use the slope and a given point to substitute for and in the point-slope form , which is derived from the slope equation .
Step 2.2
Simplify the equation and keep it in point-slope form.
Step 2.3
Solve for .
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Step 2.3.1
Add and .
Step 2.3.2
Simplify .
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Step 2.3.2.1
Apply the distributive property.
Step 2.3.2.2
Rewrite as .
Step 2.3.2.3
Multiply .
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Step 2.3.2.3.1
Multiply by .
Step 2.3.2.3.2
Multiply by .
Step 3