Calculus Examples

Find dy/dx y=(5-7x+9x^2)/(x^2)
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2
Differentiate.
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Step 3.2.1
Multiply the exponents in .
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Step 3.2.1.1
Apply the power rule and multiply exponents, .
Step 3.2.1.2
Multiply by .
Step 3.2.2
By the Sum Rule, the derivative of with respect to is .
Step 3.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.4
Add and .
Step 3.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.6
Differentiate using the Power Rule which states that is where .
Step 3.2.7
Multiply by .
Step 3.2.8
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.9
Differentiate using the Power Rule which states that is where .
Step 3.2.10
Multiply by .
Step 3.2.11
Differentiate using the Power Rule which states that is where .
Step 3.2.12
Simplify with factoring out.
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Step 3.2.12.1
Multiply by .
Step 3.2.12.2
Factor out of .
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Step 3.2.12.2.1
Factor out of .
Step 3.2.12.2.2
Factor out of .
Step 3.2.12.2.3
Factor out of .
Step 3.3
Cancel the common factors.
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Step 3.3.1
Factor out of .
Step 3.3.2
Cancel the common factor.
Step 3.3.3
Rewrite the expression.
Step 3.4
Simplify.
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Step 3.4.1
Apply the distributive property.
Step 3.4.2
Apply the distributive property.
Step 3.4.3
Simplify the numerator.
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Step 3.4.3.1
Simplify each term.
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Step 3.4.3.1.1
Move to the left of .
Step 3.4.3.1.2
Rewrite using the commutative property of multiplication.
Step 3.4.3.1.3
Multiply by by adding the exponents.
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Step 3.4.3.1.3.1
Move .
Step 3.4.3.1.3.2
Multiply by .
Step 3.4.3.1.4
Multiply by .
Step 3.4.3.1.5
Multiply by .
Step 3.4.3.1.6
Multiply by .
Step 3.4.3.2
Combine the opposite terms in .
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Step 3.4.3.2.1
Subtract from .
Step 3.4.3.2.2
Add and .
Step 3.4.3.3
Add and .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .