Calculus Examples

Find dy/dx x natural log of y+y^3 = natural log of x
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate the left side of the equation.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2.2
Differentiate using the chain rule, which states that is where and .
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Step 2.2.2.1
To apply the Chain Rule, set as .
Step 2.2.2.2
The derivative of with respect to is .
Step 2.2.2.3
Replace all occurrences of with .
Step 2.2.3
Rewrite as .
Step 2.2.4
Differentiate using the Power Rule which states that is where .
Step 2.2.5
Combine and .
Step 2.2.6
Combine and .
Step 2.2.7
Multiply by .
Step 2.3
Evaluate .
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Step 2.3.1
Differentiate using the chain rule, which states that is where and .
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Step 2.3.1.1
To apply the Chain Rule, set as .
Step 2.3.1.2
Differentiate using the Power Rule which states that is where .
Step 2.3.1.3
Replace all occurrences of with .
Step 2.3.2
Rewrite as .
Step 2.4
Reorder terms.
Step 3
The derivative of with respect to is .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Solve for .
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Step 5.1
Find the LCD of the terms in the equation.
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Step 5.1.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 5.1.2
Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
Step 5.1.3
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 5.1.4
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 5.1.5
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 5.1.6
The factor for is itself.
occurs time.
Step 5.1.7
The factor for is itself.
occurs time.
Step 5.1.8
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 5.2
Multiply each term in by to eliminate the fractions.
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Step 5.2.1
Multiply each term in by .
Step 5.2.2
Simplify the left side.
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Step 5.2.2.1
Simplify each term.
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Step 5.2.2.1.1
Multiply by by adding the exponents.
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Step 5.2.2.1.1.1
Move .
Step 5.2.2.1.1.2
Multiply by .
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Step 5.2.2.1.1.2.1
Raise to the power of .
Step 5.2.2.1.1.2.2
Use the power rule to combine exponents.
Step 5.2.2.1.1.3
Add and .
Step 5.2.2.1.2
Cancel the common factor of .
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Step 5.2.2.1.2.1
Factor out of .
Step 5.2.2.1.2.2
Cancel the common factor.
Step 5.2.2.1.2.3
Rewrite the expression.
Step 5.2.2.1.3
Raise to the power of .
Step 5.2.2.1.4
Raise to the power of .
Step 5.2.2.1.5
Use the power rule to combine exponents.
Step 5.2.2.1.6
Add and .
Step 5.2.2.2
Reorder factors in .
Step 5.2.3
Simplify the right side.
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Step 5.2.3.1
Cancel the common factor of .
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Step 5.2.3.1.1
Factor out of .
Step 5.2.3.1.2
Cancel the common factor.
Step 5.2.3.1.3
Rewrite the expression.
Step 5.3
Solve the equation.
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Step 5.3.1
Subtract from both sides of the equation.
Step 5.3.2
Factor out of .
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Step 5.3.2.1
Factor out of .
Step 5.3.2.2
Factor out of .
Step 5.3.2.3
Factor out of .
Step 5.3.3
Divide each term in by and simplify.
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Step 5.3.3.1
Divide each term in by .
Step 5.3.3.2
Simplify the left side.
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Step 5.3.3.2.1
Cancel the common factor of .
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Step 5.3.3.2.1.1
Cancel the common factor.
Step 5.3.3.2.1.2
Rewrite the expression.
Step 5.3.3.2.2
Cancel the common factor of .
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Step 5.3.3.2.2.1
Cancel the common factor.
Step 5.3.3.2.2.2
Divide by .
Step 5.3.3.3
Simplify the right side.
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Step 5.3.3.3.1
Simplify each term.
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Step 5.3.3.3.1.1
Cancel the common factor of .
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Step 5.3.3.3.1.1.1
Cancel the common factor.
Step 5.3.3.3.1.1.2
Rewrite the expression.
Step 5.3.3.3.1.2
Move the negative in front of the fraction.
Step 5.3.3.3.2
To write as a fraction with a common denominator, multiply by .
Step 5.3.3.3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 5.3.3.3.3.1
Multiply by .
Step 5.3.3.3.3.2
Reorder the factors of .
Step 5.3.3.3.4
Combine the numerators over the common denominator.
Step 5.3.3.3.5
Simplify the numerator.
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Step 5.3.3.3.5.1
Factor out of .
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Step 5.3.3.3.5.1.1
Raise to the power of .
Step 5.3.3.3.5.1.2
Factor out of .
Step 5.3.3.3.5.1.3
Factor out of .
Step 5.3.3.3.5.1.4
Factor out of .
Step 5.3.3.3.5.2
Rewrite as .
Step 5.3.3.3.6
Reorder factors in .
Step 6
Replace with .