Calculus Examples

Find the Derivative - d/dx natural log of x+ square root of 3+x^2
Step 1
Use to rewrite as .
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
The derivative of with respect to is .
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
Differentiate using the Power Rule which states that is where .
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
Differentiate using the Power Rule which states that is where .
Step 4.3
Replace all occurrences of with .
Step 5
To write as a fraction with a common denominator, multiply by .
Step 6
Combine and .
Step 7
Combine the numerators over the common denominator.
Step 8
Simplify the numerator.
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Step 8.1
Multiply by .
Step 8.2
Subtract from .
Step 9
Combine fractions.
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Step 9.1
Move the negative in front of the fraction.
Step 9.2
Combine and .
Step 9.3
Move to the denominator using the negative exponent rule .
Step 10
By the Sum Rule, the derivative of with respect to is .
Step 11
Since is constant with respect to , the derivative of with respect to is .
Step 12
Add and .
Step 13
Differentiate using the Power Rule which states that is where .
Step 14
Simplify terms.
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Step 14.1
Combine and .
Step 14.2
Combine and .
Step 14.3
Cancel the common factor.
Step 14.4
Rewrite the expression.
Step 15
Simplify.
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Step 15.1
Reorder the factors of .
Step 15.2
Multiply by .
Step 15.3
Multiply the numerator and denominator of the fraction by .
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Step 15.3.1
Multiply by .
Step 15.3.2
Combine.
Step 15.4
Apply the distributive property.
Step 15.5
Cancel the common factor of .
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Step 15.5.1
Cancel the common factor.
Step 15.5.2
Rewrite the expression.
Step 15.6
Multiply by .
Step 15.7
Simplify the denominator.
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Step 15.7.1
Multiply by by adding the exponents.
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Step 15.7.1.1
Use the power rule to combine exponents.
Step 15.7.1.2
Combine the numerators over the common denominator.
Step 15.7.1.3
Add and .
Step 15.7.1.4
Divide by .
Step 15.7.2
Simplify .
Step 15.7.3
Reorder terms.