# Calculus Examples

Find the Tangent Line at the Point y=(7x)/(x+4) , (3,3)
,
Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
Step 1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2
Differentiate using the Quotient Rule which states that is where and .
Step 1.3
Differentiate.
Step 1.3.1
Differentiate using the Power Rule which states that is where .
Step 1.3.2
Multiply by .
Step 1.3.3
By the Sum Rule, the derivative of with respect to is .
Step 1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.6
Simplify terms.
Step 1.3.6.1
Step 1.3.6.2
Multiply by .
Step 1.3.6.3
Subtract from .
Step 1.3.6.4
Step 1.3.6.5
Combine and .
Step 1.3.6.6
Multiply by .
Step 1.4
Evaluate the derivative at .
Step 1.5
Simplify.
Step 1.5.1
Simplify the denominator.
Step 1.5.1.1
Step 1.5.1.2
Raise to the power of .
Step 1.5.2
Cancel the common factor of and .
Step 1.5.2.1
Factor out of .
Step 1.5.2.2
Cancel the common factors.
Step 1.5.2.2.1
Factor out of .
Step 1.5.2.2.2
Cancel the common factor.
Step 1.5.2.2.3
Rewrite the expression.
Step 2
Plug the slope and point values into the point-slope formula and solve for .
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 .
Step 2.3.1
Simplify .
Step 2.3.1.1
Rewrite.
Step 2.3.1.2
Step 2.3.1.3
Apply the distributive property.
Step 2.3.1.4
Combine and .
Step 2.3.1.5
Multiply .
Step 2.3.1.5.1
Combine and .
Step 2.3.1.5.2
Multiply by .
Step 2.3.1.6
Move the negative in front of the fraction.
Step 2.3.2
Move all terms not containing to the right side of the equation.
Step 2.3.2.1
Add to both sides of the equation.
Step 2.3.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.3.2.3
Combine and .
Step 2.3.2.4
Combine the numerators over the common denominator.
Step 2.3.2.5
Simplify the numerator.
Step 2.3.2.5.1
Multiply by .
Step 2.3.2.5.2