Calculus Examples

Solve for x y=(e^x-e^(-x))/2
Step 1
Rewrite the equation as .
Step 2
Multiply both sides by .
Step 3
Simplify.
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Step 3.1
Simplify the left side.
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Step 3.1.1
Cancel the common factor of .
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Step 3.1.1.1
Cancel the common factor.
Step 3.1.1.2
Rewrite the expression.
Step 3.2
Simplify the right side.
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Step 3.2.1
Move to the left of .
Step 4
Solve for .
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Step 4.1
Rewrite as exponentiation.
Step 4.2
Substitute for .
Step 4.3
Rewrite the expression using the negative exponent rule .
Step 4.4
Solve for .
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Step 4.4.1
Find the LCD of the terms in the equation.
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Step 4.4.1.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 4.4.1.2
The LCM of one and any expression is the expression.
Step 4.4.2
Multiply each term in by to eliminate the fractions.
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Step 4.4.2.1
Multiply each term in by .
Step 4.4.2.2
Simplify the left side.
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Step 4.4.2.2.1
Simplify each term.
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Step 4.4.2.2.1.1
Multiply by .
Step 4.4.2.2.1.2
Cancel the common factor of .
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Step 4.4.2.2.1.2.1
Move the leading negative in into the numerator.
Step 4.4.2.2.1.2.2
Cancel the common factor.
Step 4.4.2.2.1.2.3
Rewrite the expression.
Step 4.4.3
Solve the equation.
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Step 4.4.3.1
Subtract from both sides of the equation.
Step 4.4.3.2
Use the quadratic formula to find the solutions.
Step 4.4.3.3
Substitute the values , , and into the quadratic formula and solve for .
Step 4.4.3.4
Simplify.
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Step 4.4.3.4.1
Simplify the numerator.
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Step 4.4.3.4.1.1
Apply the product rule to .
Step 4.4.3.4.1.2
Raise to the power of .
Step 4.4.3.4.1.3
Multiply .
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Step 4.4.3.4.1.3.1
Multiply by .
Step 4.4.3.4.1.3.2
Multiply by .
Step 4.4.3.4.1.4
Factor out of .
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Step 4.4.3.4.1.4.1
Factor out of .
Step 4.4.3.4.1.4.2
Factor out of .
Step 4.4.3.4.1.4.3
Factor out of .
Step 4.4.3.4.1.5
Rewrite as .
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Step 4.4.3.4.1.5.1
Rewrite as .
Step 4.4.3.4.1.5.2
Rewrite as .
Step 4.4.3.4.1.6
Pull terms out from under the radical.
Step 4.4.3.4.1.7
One to any power is one.
Step 4.4.3.4.2
Multiply by .
Step 4.4.3.4.3
Simplify .
Step 4.4.3.5
The final answer is the combination of both solutions.
Step 4.5
Substitute for in .
Step 4.6
Solve .
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Step 4.6.1
Rewrite the equation as .
Step 4.6.2
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 4.6.3
Expand the left side.
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Step 4.6.3.1
Expand by moving outside the logarithm.
Step 4.6.3.2
The natural logarithm of is .
Step 4.6.3.3
Multiply by .
Step 4.7
Substitute for in .
Step 4.8
Solve .
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Step 4.8.1
Rewrite the equation as .
Step 4.8.2
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 4.8.3
Expand the left side.
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Step 4.8.3.1
Expand by moving outside the logarithm.
Step 4.8.3.2
The natural logarithm of is .
Step 4.8.3.3
Multiply by .
Step 4.9
List the solutions that makes the equation true.