If ƒ is continuous on the closed interval [a,b] and k is any number between ƒ(a) and ƒ(b), then there is at least one number c in [a,b] such that ƒ(c)=k.

1. Differentiate both sides of the equation with respect to x.
2. Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.
3. Factor dy/dx out of the left side of the equation.
4. Solve for dy/dx by dividing both sides of the equation by the left-hand factor that does not contain dy/dx.

1. Make a sketch and label with appropriate variables
2. Write Find, When, Given
3. Write equation-involving variables whose rate of change either are given or are to be determined.
4. Implicitly differentiate both sides with respect to time.
5. After differentiating, substitute known values and solve for required rate of change.
   

1. Draw a sketch and label with appropriate variable.
2. Write a primary equation for the quantity that is to be maximized or minimized.
3. Reduce the primary equation to one having a single variable. This may involve the sum of a secondary equation.
4. Find the derivative of the equation and determine the correct value.
   

Domain:   (0,∞)

Range:     (-∞,∞)

ln(1) = 0

ln(ab) = ln(a) + ln(b)

ln(aⁿ) = n ln(a)

ln(a/b) = ln(a) - ln(b)

1. Set function equal to given value and solve
2. Find derivative of given function
3. Substitute the answer from Step 1 into the reciprocal of the derivative
   

Domain: (-∞, ∞)

Range: (0, ∞)

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e^ae^b = e^a+b

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