**Q: Find the derivative when x = 1 of the function f(x) = 1/√(x)**

# Category: Differentiation

# Average rate of change vs. Instantaneous rate of change

**Q: Let f(x) = 3x² – x**

**a. Find the average rate of change of f(x) with the respect to x as x changes from x = 0 to x = 1/16**

**b. Use calculus to find the instantaneous rate of change of f(x) at x=0 and compare with the average rate found in part (a).**

# Tangent Line

**Q: Find the derivative of function, then find the equation of the line that is tangent to its graph for the specified value x=c**

**f(x) = x²; c=1**

# Find the equation of the tangent line!

**Q: Let f(x) = -2/x . Find the equation of the tangent line when x = -1.**

# Horizontal Tangent Line

**Q: Let f(x)= 3sin ^{2}x + 4cos^{3}x. Determine where the tangent line is horizontal.**

# Quotient with chain rule

**Q: F(x) = (3x²+1) ^{3}/(x^{2}-1)^{4}. Find the derivative of F(x), F'(x).**

# Business Calculus Word Problem

**BACKGROUND**

First we need some basics (assuming everything is linear, we continue):

We logically know that:

Profit = What you make – What you spend

In math, that is:

**P = Revenue – Cost**

**(1) P = R – C**

And,

Revenue = price * quantity

**(2) R = px**

**(3) Cost** = **(variable cost)*x + (fixed cost)**

Now, there is a difference between big **P (profit) **and little **p (price or demand)**

We usually assume price is linear, so:

**(4) p = mx + b**

Everything in **BOLD** are things you must know!

OK…. Now let’s start deciphering the actual problem:

**Q: A manufacturer sells 150 tables a month at the price of $200 each. For each $1 decrease in price, he can sell 25 more tables. The tables cost $125 to make. Express monthly profit as a function of the price, draw a graph and estimate the optimal selling price.**

# Slope of the Tangent Line

**Q: Find the point(s) on the graph of the function f(x) = (x² + 6)(x – 5) where the slope of the tangent line is -2.**