Calculus Examples

Find the Tangent Line at (3,-4) x^2+y^2=25 , (3,-4)
,
Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
Tap for more steps...
Step 1.1
Differentiate both sides of the equation.
Step 1.2
Differentiate the left side of the equation.
Tap for more steps...
Step 1.2.1
Differentiate.
Tap for more steps...
Step 1.2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.1.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2
Evaluate .
Tap for more steps...
Step 1.2.2.1
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 1.2.2.1.1
To apply the Chain Rule, set as .
Step 1.2.2.1.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2.1.3
Replace all occurrences of with .
Step 1.2.2.2
Rewrite as .
Step 1.2.3
Reorder terms.
Step 1.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.4
Reform the equation by setting the left side equal to the right side.
Step 1.5
Solve for .
Tap for more steps...
Step 1.5.1
Subtract from both sides of the equation.
Step 1.5.2
Divide each term in by and simplify.
Tap for more steps...
Step 1.5.2.1
Divide each term in by .
Step 1.5.2.2
Simplify the left side.
Tap for more steps...
Step 1.5.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 1.5.2.2.1.1
Cancel the common factor.
Step 1.5.2.2.1.2
Rewrite the expression.
Step 1.5.2.2.2
Cancel the common factor of .
Tap for more steps...
Step 1.5.2.2.2.1
Cancel the common factor.
Step 1.5.2.2.2.2
Divide by .
Step 1.5.2.3
Simplify the right side.
Tap for more steps...
Step 1.5.2.3.1
Cancel the common factor of and .
Tap for more steps...
Step 1.5.2.3.1.1
Factor out of .
Step 1.5.2.3.1.2
Cancel the common factors.
Tap for more steps...
Step 1.5.2.3.1.2.1
Factor out of .
Step 1.5.2.3.1.2.2
Cancel the common factor.
Step 1.5.2.3.1.2.3
Rewrite the expression.
Step 1.5.2.3.2
Move the negative in front of the fraction.
Step 1.6
Replace with .
Step 1.7
Evaluate at and .
Tap for more steps...
Step 1.7.1
Replace the variable with in the expression.
Step 1.7.2
Replace the variable with in the expression.
Step 1.7.3
Move the negative in front of the fraction.
Step 1.7.4
Multiply .
Tap for more steps...
Step 1.7.4.1
Multiply by .
Step 1.7.4.2
Multiply by .
Step 2
Plug the slope and point values into the point-slope formula and solve for .
Tap for more steps...
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 .
Tap for more steps...
Step 2.3.1
Simplify .
Tap for more steps...
Step 2.3.1.1
Rewrite.
Step 2.3.1.2
Simplify by adding zeros.
Step 2.3.1.3
Apply the distributive property.
Step 2.3.1.4
Combine and .
Step 2.3.1.5
Multiply .
Tap for more steps...
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.
Tap for more steps...
Step 2.3.2.1
Subtract from 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.
Tap for more steps...
Step 2.3.2.5.1
Multiply by .
Step 2.3.2.5.2
Subtract from .
Step 2.3.2.6
Move the negative in front of the fraction.
Step 2.3.3
Reorder terms.
Step 3