Calculus Examples

Find dy/dx y=x^2 log base 2 of 5-4x
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 Product Rule which states that is where and .
Step 3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate.
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Step 3.3.1
Combine and .
Step 3.3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.4
Add and .
Step 3.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.6
Combine fractions.
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Step 3.3.6.1
Combine and .
Step 3.3.6.2
Move the negative in front of the fraction.
Step 3.3.7
Differentiate using the Power Rule which states that is where .
Step 3.3.8
Multiply by .
Step 3.3.9
Differentiate using the Power Rule which states that is where .
Step 3.3.10
Move .
Step 3.4
To write as a fraction with a common denominator, multiply by .
Step 3.5
Combine and .
Step 3.6
Combine the numerators over the common denominator.
Step 3.7
Simplify.
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Step 3.7.1
Apply the distributive property.
Step 3.7.2
Apply the distributive property.
Step 3.7.3
Simplify the numerator.
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Step 3.7.3.1
Simplify each term.
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Step 3.7.3.1.1
Simplify by moving inside the logarithm.
Step 3.7.3.1.2
Simplify each term.
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Step 3.7.3.1.2.1
Simplify by moving inside the logarithm.
Step 3.7.3.1.2.2
Raise to the power of .
Step 3.7.3.1.2.3
Simplify by moving inside the logarithm.
Step 3.7.3.1.2.4
Raise to the power of .
Step 3.7.3.1.3
Apply the distributive property.
Step 3.7.3.1.4
Rewrite using the commutative property of multiplication.
Step 3.7.3.1.5
Multiply by by adding the exponents.
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Step 3.7.3.1.5.1
Move .
Step 3.7.3.1.5.2
Multiply by .
Step 3.7.3.2
Reorder factors in .
Step 3.7.4
Factor out of .
Step 3.7.5
Factor out of .
Step 3.7.6
Factor out of .
Step 3.7.7
Factor out of .
Step 3.7.8
Factor out of .
Step 3.7.9
Rewrite as .
Step 3.7.10
Factor out of .
Step 3.7.11
Factor out of .
Step 3.7.12
Factor out of .
Step 3.7.13
Rewrite as .
Step 3.7.14
Cancel the common factor.
Step 3.7.15
Rewrite the expression.
Step 3.7.16
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .