Calculus Examples

Find the Tangent Line at the Point y = square root of 2x , (18,6)
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Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
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Step 1.1
Simplify with factoring out.
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Step 1.1.1
Use to rewrite as .
Step 1.1.2
Factor out of .
Step 1.1.3
Apply the product rule to .
Step 1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.3
Differentiate using the Power Rule which states that is where .
Step 1.4
To write as a fraction with a common denominator, multiply by .
Step 1.5
Combine and .
Step 1.6
Combine the numerators over the common denominator.
Step 1.7
Simplify the numerator.
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Step 1.7.1
Multiply by .
Step 1.7.2
Subtract from .
Step 1.8
Move the negative in front of the fraction.
Step 1.9
Combine and .
Step 1.10
Combine and .
Step 1.11
Simplify the expression.
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Step 1.11.1
Move to the denominator using the negative exponent rule .
Step 1.11.2
Move to the denominator using the negative exponent rule .
Step 1.12
Multiply by by adding the exponents.
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Step 1.12.1
Multiply by .
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Step 1.12.1.1
Raise to the power of .
Step 1.12.1.2
Use the power rule to combine exponents.
Step 1.12.2
Write as a fraction with a common denominator.
Step 1.12.3
Combine the numerators over the common denominator.
Step 1.12.4
Subtract from .
Step 1.13
Evaluate the derivative at .
Step 1.14
Remove parentheses.
Step 2
Plug the slope and point values into the point-slope formula and solve for .
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Step 2.1
Use the slope and a given point to substitute for and in the point-slope form , which is derived from the slope equation .
Step 2.2
Simplify the equation and keep it in point-slope form.
Step 2.3
Solve for .
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Step 2.3.1
Simplify .
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Step 2.3.1.1
Rewrite.
Step 2.3.1.2
Simplify terms.
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Step 2.3.1.2.1
Apply the distributive property.
Step 2.3.1.2.2
Combine and .
Step 2.3.1.2.3
Combine and .
Step 2.3.1.3
Simplify each term.
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Step 2.3.1.3.1
Move the negative in front of the fraction.
Step 2.3.1.3.2
Move to the numerator using the negative exponent rule .
Step 2.3.1.3.3
Multiply by by adding the exponents.
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Step 2.3.1.3.3.1
Multiply by .
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Step 2.3.1.3.3.1.1
Raise to the power of .
Step 2.3.1.3.3.1.2
Use the power rule to combine exponents.
Step 2.3.1.3.3.2
Write as a fraction with a common denominator.
Step 2.3.1.3.3.3
Combine the numerators over the common denominator.
Step 2.3.1.3.3.4
Subtract from .
Step 2.3.1.3.4
Use the power of quotient rule .
Step 2.3.1.3.5
Divide by .
Step 2.3.1.3.6
Rewrite as .
Step 2.3.1.3.7
Apply the power rule and multiply exponents, .
Step 2.3.1.3.8
Cancel the common factor of .
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Step 2.3.1.3.8.1
Cancel the common factor.
Step 2.3.1.3.8.2
Rewrite the expression.
Step 2.3.1.3.9
Evaluate the exponent.
Step 2.3.1.3.10
Multiply by .
Step 2.3.2
Move all terms not containing to the right side of the equation.
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Step 2.3.2.1
Add to both sides of the equation.
Step 2.3.2.2
Add and .
Step 2.3.3
Reorder terms.
Step 3