# Derivatives

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Finding Derivatives with Limits

How to find the equation of a tangent line

Linear Approximation (Linearization) and Differentials

L’Hospital’s Rule

Examples of product, quotient, and chain rules

Solution – Find two numbers whose sum is 23 and whose product is a maximum

Solution – Find the values of a and b that make the function differentiable everywhere

Solution – Find $\frac{dy}{dx}$ if $y=x^x$

Solution – Find dy/dx by implicit differentiation examples

#### Related rates problems:

Triangles

Solution – At noon, ship A is 150 km west of ship B.  Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h.  How fast is the distance between the ships changing at 4:00 PM?

Solution – A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

Solution – A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is $\frac{\pi}{3}$, this angle is decreasing at a rate of $\frac{\pi}{6} \ \frac{rad}{min}$. How fast is the plane traveling at this time?

Solution – The top of a ladder slides down a vertical wall at a rate of 0.15 $\frac{m}{s}$. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 $\frac{m}{s}$. How long is the ladder?

Squares

Solution – Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 $cm^2$?

Cones

Solution – Water is leaking out of an inverted conical tank at a rate of 10,000 $\frac{cm^3}{min}$ at the same time water is being pumped into the tank at a constant rate. The tank has a height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 $\frac{cm}{min}$ when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

Spheres

Solution – If a snowball melts so that its surface area decreases at a rate of 1 $\frac{cm^2}{min}$, find the rate at which the diameter decreases when the diameter is 10 cm.

Solution – The radius of a sphere is increasing at a rate of 4 $\frac{mm}{s}$. How fast is the volume increasing when the diameter is 80 mm?

Cylinders

Solution – A cylindrical tank with radius 5 m is being ﬁlled with water at a rate of 3 $\frac{m^3}{min}$. How fast is the height of the water increasing?