Calculus Examples

Find the Derivative - d/dt (3 natural log of t)/(at-bt^2)
Step 1
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
The derivative of with respect to is .
Step 4
Differentiate.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.3
Differentiate using the Power Rule which states that is where .
Step 4.4
Multiply by .
Step 4.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.6
Differentiate using the Power Rule which states that is where .
Step 4.7
Combine fractions.
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Step 4.7.1
Multiply by .
Step 4.7.2
Combine and .
Step 5
Simplify.
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Step 5.1
Apply the distributive property.
Step 5.2
Apply the distributive property.
Step 5.3
Apply the distributive property.
Step 5.4
Simplify the numerator.
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Step 5.4.1
Simplify each term.
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Step 5.4.1.1
Cancel the common factor of .
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Step 5.4.1.1.1
Factor out of .
Step 5.4.1.1.2
Cancel the common factor.
Step 5.4.1.1.3
Rewrite the expression.
Step 5.4.1.2
Cancel the common factor of .
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Step 5.4.1.2.1
Factor out of .
Step 5.4.1.2.2
Cancel the common factor.
Step 5.4.1.2.3
Rewrite the expression.
Step 5.4.1.3
Multiply by .
Step 5.4.1.4
Multiply .
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Step 5.4.1.4.1
Multiply by .
Step 5.4.1.4.2
Reorder and .
Step 5.4.1.4.3
Simplify by moving inside the logarithm.
Step 5.4.1.5
Multiply .
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Step 5.4.1.5.1
Multiply by .
Step 5.4.1.5.2
Simplify by moving inside the logarithm.
Step 5.4.1.6
Multiply .
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Step 5.4.1.6.1
Reorder and .
Step 5.4.1.6.2
Simplify by moving inside the logarithm.
Step 5.4.1.7
Multiply the exponents in .
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Step 5.4.1.7.1
Apply the power rule and multiply exponents, .
Step 5.4.1.7.2
Multiply by .
Step 5.4.2
Reorder factors in .
Step 5.5
Reorder terms.
Step 5.6
Simplify the denominator.
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Step 5.6.1
Factor out of .
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Step 5.6.1.1
Factor out of .
Step 5.6.1.2
Factor out of .
Step 5.6.1.3
Factor out of .
Step 5.6.2
Apply the product rule to .