# 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.
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Quotient Rule which states that is where and .
Differentiate.
Differentiate using the Power Rule which states that is where .
Multiply by .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Simplify terms.
Multiply by .
Subtract from .
Combine and .
Multiply by .
Evaluate the derivative at .
Simplify.
Simplify the denominator.
Raise to the power of .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Step 2
Plug the slope and point values into the point-slope formula and solve for .
Use the slope and a given point to substitute for and in the point-slope form , which is derived from the slope equation .
Simplify the equation and keep it in point-slope form.
Solve for .
Simplify .
Rewrite.
Apply the distributive property.
Combine and .
Multiply .
Combine and .
Multiply by .
Move the negative in front of the fraction.
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .