Calculus Examples

Find dx/dy y=1/(3x^3)+(x^7)/10
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate using the Power Rule which states that is where .
Step 3
Differentiate the right side of the equation.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
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Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Differentiate using the Quotient Rule which states that is where and .
Step 3.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.4
Differentiate using the chain rule, which states that is where and .
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Step 3.2.4.1
To apply the Chain Rule, set as .
Step 3.2.4.2
Differentiate using the Power Rule which states that is where .
Step 3.2.4.3
Replace all occurrences of with .
Step 3.2.5
Rewrite as .
Step 3.2.6
Multiply by .
Step 3.2.7
Multiply by .
Step 3.2.8
Multiply by .
Step 3.2.9
Subtract from .
Step 3.2.10
Multiply the exponents in .
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Step 3.2.10.1
Apply the power rule and multiply exponents, .
Step 3.2.10.2
Multiply by .
Step 3.2.11
Cancel the common factor of and .
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Step 3.2.11.1
Factor out of .
Step 3.2.11.2
Cancel the common factors.
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Step 3.2.11.2.1
Factor out of .
Step 3.2.11.2.2
Cancel the common factor.
Step 3.2.11.2.3
Rewrite the expression.
Step 3.2.12
Move the negative in front of the fraction.
Step 3.2.13
Multiply by .
Step 3.2.14
Cancel the common factor of .
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Step 3.2.14.1
Cancel the common factor.
Step 3.2.14.2
Rewrite the expression.
Step 3.3
Evaluate .
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Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.3.2.1
To apply the Chain Rule, set as .
Step 3.3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.3.2.3
Replace all occurrences of with .
Step 3.3.3
Rewrite as .
Step 3.3.4
Combine and .
Step 3.3.5
Combine and .
Step 3.3.6
Combine and .
Step 3.3.7
Move to the left of .
Step 3.4
Reorder terms.
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
Rewrite the equation as .
Step 5.2
Find the LCD of the terms in the equation.
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Step 5.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 5.2.2
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 5.2.3
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 5.2.4
has factors of and .
Step 5.2.5
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 5.2.6
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 5.2.7
Multiply by .
Step 5.2.8
The factors for are , which is multiplied by each other times.
occurs times.
Step 5.2.9
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 5.2.10
Simplify .
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Step 5.2.10.1
Multiply by .
Step 5.2.10.2
Multiply by by adding the exponents.
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Step 5.2.10.2.1
Multiply by .
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Step 5.2.10.2.1.1
Raise to the power of .
Step 5.2.10.2.1.2
Use the power rule to combine exponents.
Step 5.2.10.2.2
Add and .
Step 5.2.10.3
Multiply by by adding the exponents.
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Step 5.2.10.3.1
Multiply by .
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Step 5.2.10.3.1.1
Raise to the power of .
Step 5.2.10.3.1.2
Use the power rule to combine exponents.
Step 5.2.10.3.2
Add and .
Step 5.2.11
The LCM for is the numeric part multiplied by the variable part.
Step 5.3
Multiply each term in by to eliminate the fractions.
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Step 5.3.1
Multiply each term in by .
Step 5.3.2
Simplify the left side.
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Step 5.3.2.1
Simplify each term.
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Step 5.3.2.1.1
Rewrite using the commutative property of multiplication.
Step 5.3.2.1.2
Cancel the common factor of .
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Step 5.3.2.1.2.1
Cancel the common factor.
Step 5.3.2.1.2.2
Rewrite the expression.
Step 5.3.2.1.3
Multiply by by adding the exponents.
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Step 5.3.2.1.3.1
Move .
Step 5.3.2.1.3.2
Use the power rule to combine exponents.
Step 5.3.2.1.3.3
Add and .
Step 5.3.2.1.4
Cancel the common factor of .
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Step 5.3.2.1.4.1
Move the leading negative in into the numerator.
Step 5.3.2.1.4.2
Factor out of .
Step 5.3.2.1.4.3
Cancel the common factor.
Step 5.3.2.1.4.4
Rewrite the expression.
Step 5.3.2.1.5
Multiply by .
Step 5.3.3
Simplify the right side.
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Step 5.3.3.1
Multiply by .
Step 5.4
Solve the equation.
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Step 5.4.1
Factor out of .
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Step 5.4.1.1
Factor out of .
Step 5.4.1.2
Factor out of .
Step 5.4.1.3
Factor out of .
Step 5.4.2
Divide each term in by and simplify.
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Step 5.4.2.1
Divide each term in by .
Step 5.4.2.2
Simplify the left side.
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Step 5.4.2.2.1
Cancel the common factor of .
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Step 5.4.2.2.1.1
Cancel the common factor.
Step 5.4.2.2.1.2
Divide by .
Step 6
Replace with .