Calculus Examples

Find the Derivative - d/d@VAR f(x)=(e^(7x^2-3))/( natural log of 3x+5)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Differentiate using the Power Rule which states that is where .
Step 3.4
Multiply by .
Step 3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.6
Add and .
Step 4
Differentiate using the chain rule, which states that is where and .
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Step 4.1
To apply the Chain Rule, set as .
Step 4.2
The derivative of with respect to is .
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate.
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Step 5.1
Combine and .
Step 5.2
By the Sum Rule, the derivative of with respect to is .
Step 5.3
Since is constant with respect to , the derivative of with respect to is .
Step 5.4
Differentiate using the Power Rule which states that is where .
Step 5.5
Multiply by .
Step 5.6
Since is constant with respect to , the derivative of with respect to is .
Step 5.7
Combine fractions.
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Step 5.7.1
Add and .
Step 5.7.2
Multiply by .
Step 5.7.3
Combine and .
Step 5.7.4
Move the negative in front of the fraction.
Step 6
To write as a fraction with a common denominator, multiply by .
Step 7
Combine the numerators over the common denominator.
Step 8
Rewrite as a product.
Step 9
Multiply by .
Step 10
Simplify.
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Step 10.1
Simplify the numerator.
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Step 10.1.1
Simplify each term.
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Step 10.1.1.1
Rewrite using the commutative property of multiplication.
Step 10.1.1.2
Simplify by moving inside the logarithm.
Step 10.1.1.3
Apply the distributive property.
Step 10.1.1.4
Rewrite using the commutative property of multiplication.
Step 10.1.1.5
Multiply .
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Step 10.1.1.5.1
Reorder and .
Step 10.1.1.5.2
Simplify by moving inside the logarithm.
Step 10.1.1.6
Simplify each term.
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Step 10.1.1.6.1
Multiply by by adding the exponents.
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Step 10.1.1.6.1.1
Move .
Step 10.1.1.6.1.2
Multiply by .
Step 10.1.1.6.2
Simplify by moving inside the logarithm.
Step 10.1.1.6.3
Multiply the exponents in .
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Step 10.1.1.6.3.1
Apply the power rule and multiply exponents, .
Step 10.1.1.6.3.2
Multiply by .
Step 10.1.1.6.4
Multiply the exponents in .
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Step 10.1.1.6.4.1
Apply the power rule and multiply exponents, .
Step 10.1.1.6.4.2
Multiply by .
Step 10.1.2
Reorder factors in .
Step 10.2
Reorder terms.