Expert: Paul Klarreich - 9/12/2007

Question
Hello and good afternoon. I am having some problems with the following questions of derivatives. If you can please show me step by step how to solve the questions that are incorrect or don't have an answer, I will really appriciate it. Thank you so much.

1. If y = x^3 + 3x^2 + 5, what is dy/dx?

  Ans. 3x^2 + 6x +5

2. What is the derivative of y= sqrt (3x)

  Ans. dy/dx = 2 sqrt (3x)/ 3x

3. Find the derivative with respect to x of y= (3x + x^2)^5

  Ans. 5(3x +x^2)^5(3 + 2x)

4. What is the dy/dx if y- (x^2 + 2)^3(x^2 + 3)^2


5. If y = [(3x^2 + 2)/(2x^2 - 3)] Calculate dy/dx


6. What is the equation of the tangent line to the curve y= 6x + x^2 - 5 at the point (4,3)?  

Answer
Questioner:   Sarah
Category:  Calculus
Private:  No
 
Subject:  Calculus- functions and use
Question:  Hello and good afternoon. I am having some problems with the following questions of derivatives. If you can please show me step by step how to solve the questions that are incorrect or don't have an answer, I will really appreciate it. Thank you so much.
......................................
Hi, Sarah,

This is a lot of questions at once, but fortunately, most are short enough.

1. If y = x^3 + 3x^2 + 5, what is dy/dx?

 Ans. 3x^2 + 6x +5
NO: The derivative of a constant term is zero. dy/dx = 3x^2 + 6x

2. What is the derivative of y= sqrt (3x)

 Ans. dy/dx = 2 sqrt (3x)/ 3x

Write  y = (3x)^1/2 and use the chain rule:
dy/dx = (1/2)(3x)^-1/2(3) = 3/[2 (3x)^1/2 ]

3. Find the derivative with respect to x of y= (3x + x^2)^5

 Ans. 5(3x +x^2)^5(3 + 2x)
No:    5(3x +x^2)^4(3 + 2x)

4. What is the dy/dx if y = (x^2 + 2)^3(x^2 + 3)^2

This one takes a while, so here's how it goes:

First, it's a product, so you will use the product rule.
Then, for each derivative, you will use the chain rule, because each factor is a power of a function.

dy/dx = ((x^2 + 2)^3)(2(x^2 + 3)(2x)) + (3(x^2 + 2)^2(2x))(x^2 + 3)^2)

Now take out all common factors and simplify:

dy/dx = (x^2 + 2)^2(x^2 + 3)((x^2 + 2)(2(2x)) + (3(2x))(x^2 + 3))

dy/dx = (x^2 + 2)^2(x^2 + 3)(4x^3 + 8x + 6x^3 + 9x)

dy/dx = (x^2 + 2)^2(x^2 + 3)(10x^3 + 17x)



5. If y = [(3x^2 + 2)/(2x^2 - 3)] Calculate dy/dx

This is a quotient rule example.  My suggestion: Learn to love the quotient rule. It really isn't that hard.  You start by writing this pattern:
       ()() - ()()
dy/dx = ------------
           ()^2

Then fill in the proper derivatives and functions:

       (2x^2 - 3)(6x) - (3x^2 + 2)(4x)
dy/dx = -------------------------------
           (2x^2 - 3)^2

Then you just remove parentheses and simplify:

       12x^3 - 18x - 12x^3 - 8x
dy/dx = ---------------------------
           (2x^2 - 3)^2

         - 26x
dy/dx = -------------
       (2x^2 - 3)^2

6. What is the equation of the tangent line to the curve y= 6x + x^2 - 5 at the point (4,3)?

I think you can handle this:

Step 1: Confirm that (4,3) really is on the graph. Sometimes those textbook writers get careless.
Substitute  x = 3, compute y.  If it doesn't come out to 3, scream!

Step 2: Compute dy/dx, then substitute  x = 4.  Call the answer m.

Step 3. Use the point-slope form of the equation of a straight line:

       y - y0 = m(x - x0)

and substitute  (4,3) as the (x0,y0), and m as above.  Simplify.