Calculus Examples

Find the Horizontal Tangent Line y(x)=(6x)/((x-9)^2)
Step 1
Find the derivative.
Tap for more steps...
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 using the Power Rule.
Tap for more steps...
Step 1.3.1
Multiply the exponents in .
Tap for more steps...
Step 1.3.1.1
Apply the power rule and multiply exponents, .
Step 1.3.1.2
Multiply by .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 1.4
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 1.4.1
To apply the Chain Rule, set as .
Step 1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.4.3
Replace all occurrences of with .
Step 1.5
Simplify with factoring out.
Tap for more steps...
Step 1.5.1
Multiply by .
Step 1.5.2
Factor out of .
Tap for more steps...
Step 1.5.2.1
Factor out of .
Step 1.5.2.2
Factor out of .
Step 1.5.2.3
Factor out of .
Step 1.6
Cancel the common factors.
Tap for more steps...
Step 1.6.1
Factor out of .
Step 1.6.2
Cancel the common factor.
Step 1.6.3
Rewrite the expression.
Step 1.7
By the Sum Rule, the derivative of with respect to is .
Step 1.8
Differentiate using the Power Rule which states that is where .
Step 1.9
Since is constant with respect to , the derivative of with respect to is .
Step 1.10
Simplify terms.
Tap for more steps...
Step 1.10.1
Add and .
Step 1.10.2
Multiply by .
Step 1.10.3
Subtract from .
Step 1.10.4
Combine and .
Step 1.11
Simplify.
Tap for more steps...
Step 1.11.1
Apply the distributive property.
Step 1.11.2
Simplify each term.
Tap for more steps...
Step 1.11.2.1
Multiply by .
Step 1.11.2.2
Multiply by .
Step 1.11.3
Factor out of .
Tap for more steps...
Step 1.11.3.1
Factor out of .
Step 1.11.3.2
Factor out of .
Step 1.11.3.3
Factor out of .
Step 1.11.4
Factor out of .
Step 1.11.5
Rewrite as .
Step 1.11.6
Factor out of .
Step 1.11.7
Rewrite as .
Step 1.11.8
Move the negative in front of the fraction.
Step 2
Set the derivative equal to then solve the equation .
Tap for more steps...
Step 2.1
Set the numerator equal to zero.
Step 2.2
Solve the equation for .
Tap for more steps...
Step 2.2.1
Divide each term in by and simplify.
Tap for more steps...
Step 2.2.1.1
Divide each term in by .
Step 2.2.1.2
Simplify the left side.
Tap for more steps...
Step 2.2.1.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.2.1.2.1.1
Cancel the common factor.
Step 2.2.1.2.1.2
Divide by .
Step 2.2.1.3
Simplify the right side.
Tap for more steps...
Step 2.2.1.3.1
Divide by .
Step 2.2.2
Subtract from both sides of the equation.
Step 3
Solve the original function at .
Tap for more steps...
Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
Tap for more steps...
Step 3.2.1
Multiply by .
Step 3.2.2
Simplify the denominator.
Tap for more steps...
Step 3.2.2.1
Subtract from .
Step 3.2.2.2
Raise to the power of .
Step 3.2.3
Reduce the expression by cancelling the common factors.
Tap for more steps...
Step 3.2.3.1
Cancel the common factor of and .
Tap for more steps...
Step 3.2.3.1.1
Factor out of .
Step 3.2.3.1.2
Cancel the common factors.
Tap for more steps...
Step 3.2.3.1.2.1
Factor out of .
Step 3.2.3.1.2.2
Cancel the common factor.
Step 3.2.3.1.2.3
Rewrite the expression.
Step 3.2.3.2
Move the negative in front of the fraction.
Step 3.2.4
The final answer is .
Step 4
The horizontal tangent line on function is .
Step 5