Calculus Examples

Find the Derivative - d/dt (t- square root of t)/(t^(1/9))
Step 1
Use to rewrite as .
Step 2
Factor out of .
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Step 2.1
Raise to the power of .
Step 2.2
Factor out of .
Step 2.3
Factor out of .
Step 2.4
Factor out of .
Step 3
Move to the numerator using the negative exponent rule .
Step 4
Multiply by by adding the exponents.
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Step 4.1
Move .
Step 4.2
Use the power rule to combine exponents.
Step 4.3
To write as a fraction with a common denominator, multiply by .
Step 4.4
To write as a fraction with a common denominator, multiply by .
Step 4.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.5.1
Multiply by .
Step 4.5.2
Multiply by .
Step 4.5.3
Multiply by .
Step 4.5.4
Multiply by .
Step 4.6
Combine the numerators over the common denominator.
Step 4.7
Add and .
Step 5
Differentiate using the Product Rule which states that is where and .
Step 6
Differentiate.
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Step 6.1
By the Sum Rule, the derivative of with respect to is .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 7
To write as a fraction with a common denominator, multiply by .
Step 8
Combine and .
Step 9
Combine the numerators over the common denominator.
Step 10
Simplify the numerator.
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Step 10.1
Multiply by .
Step 10.2
Subtract from .
Step 11
Combine fractions.
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Step 11.1
Move the negative in front of the fraction.
Step 11.2
Combine and .
Step 11.3
Move to the denominator using the negative exponent rule .
Step 12
Since is constant with respect to , the derivative of with respect to is .
Step 13
Combine fractions.
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Step 13.1
Add and .
Step 13.2
Combine and .
Step 13.3
Move to the denominator using the negative exponent rule .
Step 14
Multiply by by adding the exponents.
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Step 14.1
Move .
Step 14.2
Use the power rule to combine exponents.
Step 14.3
To write as a fraction with a common denominator, multiply by .
Step 14.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 14.4.1
Multiply by .
Step 14.4.2
Multiply by .
Step 14.5
Combine the numerators over the common denominator.
Step 14.6
Add and .
Step 14.7
Cancel the common factor of and .
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Step 14.7.1
Factor out of .
Step 14.7.2
Cancel the common factors.
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Step 14.7.2.1
Factor out of .
Step 14.7.2.2
Cancel the common factor.
Step 14.7.2.3
Rewrite the expression.
Step 15
Differentiate using the Power Rule which states that is where .
Step 16
To write as a fraction with a common denominator, multiply by .
Step 17
Combine and .
Step 18
Combine the numerators over the common denominator.
Step 19
Simplify the numerator.
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Step 19.1
Multiply by .
Step 19.2
Subtract from .
Step 20
Move the negative in front of the fraction.
Step 21
Combine and .
Step 22
Move to the denominator using the negative exponent rule .
Step 23
To write as a fraction with a common denominator, multiply by .
Step 24
Combine and .
Step 25
Combine the numerators over the common denominator.
Step 26
Combine and .
Step 27
Multiply by .
Step 28
Combine and .
Step 29
Simplify the expression.
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Step 29.1
Move to the left of .
Step 29.2
Move to the denominator using the negative exponent rule .
Step 30
Multiply by by adding the exponents.
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Step 30.1
Move .
Step 30.2
Use the power rule to combine exponents.
Step 30.3
To write as a fraction with a common denominator, multiply by .
Step 30.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 30.4.1
Multiply by .
Step 30.4.2
Multiply by .
Step 30.5
Combine the numerators over the common denominator.
Step 30.6
Add and .
Step 30.7
Cancel the common factor of and .
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Step 30.7.1
Factor out of .
Step 30.7.2
Cancel the common factors.
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Step 30.7.2.1
Factor out of .
Step 30.7.2.2
Cancel the common factor.
Step 30.7.2.3
Rewrite the expression.
Step 31
Factor out of .
Step 32
Cancel the common factors.
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Step 32.1
Factor out of .
Step 32.2
Cancel the common factor.
Step 32.3
Rewrite the expression.
Step 33
Simplify.
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Step 33.1
Apply the distributive property.
Step 33.2
Simplify the numerator.
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Step 33.2.1
Simplify each term.
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Step 33.2.1.1
Cancel the common factor of .
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Step 33.2.1.1.1
Factor out of .
Step 33.2.1.1.2
Cancel the common factor.
Step 33.2.1.1.3
Rewrite the expression.
Step 33.2.1.2
Rewrite as .
Step 33.2.2
Write as a fraction with a common denominator.
Step 33.2.3
Combine the numerators over the common denominator.
Step 33.2.4
Add and .
Step 33.3
Combine terms.
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Step 33.3.1
Multiply by .
Step 33.3.2
Combine.
Step 33.3.3
Apply the distributive property.
Step 33.3.4
Cancel the common factor of .
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Step 33.3.4.1
Cancel the common factor.
Step 33.3.4.2
Rewrite the expression.
Step 33.3.5
Multiply by .
Step 33.3.6
Combine and .
Step 33.3.7
Multiply by .
Step 33.3.8
Factor out of .
Step 33.3.9
Cancel the common factors.
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Step 33.3.9.1
Factor out of .
Step 33.3.9.2
Cancel the common factor.
Step 33.3.9.3
Rewrite the expression.
Step 33.3.10
Move the negative in front of the fraction.
Step 33.3.11
Multiply by .
Step 33.4
Simplify the numerator.
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Step 33.4.1
To write as a fraction with a common denominator, multiply by .
Step 33.4.2
Combine the numerators over the common denominator.
Step 33.5
Multiply the numerator by the reciprocal of the denominator.
Step 33.6
Multiply .
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Step 33.6.1
Multiply by .
Step 33.6.2
Multiply by by adding the exponents.
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Step 33.6.2.1
Move .
Step 33.6.2.2
Use the power rule to combine exponents.
Step 33.6.2.3
To write as a fraction with a common denominator, multiply by .
Step 33.6.2.4
To write as a fraction with a common denominator, multiply by .
Step 33.6.2.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 33.6.2.5.1
Multiply by .
Step 33.6.2.5.2
Multiply by .
Step 33.6.2.5.3
Multiply by .
Step 33.6.2.5.4
Multiply by .
Step 33.6.2.6
Combine the numerators over the common denominator.
Step 33.6.2.7
Add and .
Step 33.7
Move to the left of .