Calculus Examples

Find the Tangent Line at (π/2,6π) y=12xsin(x) , (pi/2,6pi)
,
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
Since is constant with respect to , the derivative of with respect to is .
Step 1.2
Differentiate using the Product Rule which states that is where and .
Step 1.3
The derivative of with respect to is .
Step 1.4
Differentiate using the Power Rule.
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Step 1.4.1
Differentiate using the Power Rule which states that is where .
Step 1.4.2
Multiply by .
Step 1.5
Apply the distributive property.
Step 1.6
Evaluate the derivative at .
Step 1.7
Simplify.
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Step 1.7.1
Simplify each term.
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Step 1.7.1.1
Cancel the common factor of .
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Step 1.7.1.1.1
Factor out of .
Step 1.7.1.1.2
Cancel the common factor.
Step 1.7.1.1.3
Rewrite the expression.
Step 1.7.1.2
The exact value of is .
Step 1.7.1.3
Multiply .
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Step 1.7.1.3.1
Multiply by .
Step 1.7.1.3.2
Multiply by .
Step 1.7.1.4
The exact value of is .
Step 1.7.1.5
Multiply by .
Step 1.7.2
Add and .
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
Simplify .
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Step 2.3.1.1
Rewrite.
Step 2.3.1.2
Simplify by adding zeros.
Step 2.3.1.3
Apply the distributive property.
Step 2.3.1.4
Cancel the common factor of .
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Step 2.3.1.4.1
Move the leading negative in into the numerator.
Step 2.3.1.4.2
Factor out of .
Step 2.3.1.4.3
Cancel the common factor.
Step 2.3.1.4.4
Rewrite the expression.
Step 2.3.1.5
Multiply by .
Step 2.3.2
Move all terms not containing to the right side of the equation.
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Step 2.3.2.1
Add to both sides of the equation.
Step 2.3.2.2
Combine the opposite terms in .
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Step 2.3.2.2.1
Add and .
Step 2.3.2.2.2
Add and .
Step 3