Use the alternative form of the derivative to find the derivative of the function

Question

1.Use the alternative form of the derivative to find the derivative of the function f(x)=3/x^2

 

2.Find an equation of the tangent line to the graph of the function f(x)=sqrt x-2 at point (18,4)

 

3. Find the derivative of the function sinx/ 2x+cosx . Simplify your answer.

 

4.Find the derivative of the function f(fata)=7/5 sin^2 2fata.

Answer 1

1)

 

Use the limit definition of derivative.

 

lim      f(x + h) - f(x)

h->0   ____________

                    h

 

 

I think you can take it from here.

 

 

2)

 

Find the derivative of f(x) using the chain rule.

 

f'(x) = (1/2)(x - 2)^-1/2

 

f'(x) = 1 / [2√(x - 2)]

 

Evaluate the derivative at x=18.  This will be the slope of the tangent line.

 

f'(18) = 1 / (2√16)

         = 1 / 8

 

 

Next, use the point slope form of the line.

 

y = mx - mx₁ + y₁

 

where:

m = slope of tangent line

(x₁ , y₁) is the point of tangency

 

y = (1/8)x - (1/8)(18) + 4

y = (1/8)x - (9/4) + 4

y = (1/8)x - (7/4)

 

 

3)

 

You should use the quotient rule for this one.

 

 

4)

 

You should use the product rule for this one.

 

 

Alternatives for questions 3 and 4:   You can use the limit definition of derivative.  But since you did you clarify your functions with parentheses, I cannot show you how.