3.4 Product and Quotient Rules

Question 1

What is the Product Rule of Differentiation

Answer 1

If f(x) and g(x) are differentiable functions then it would be
d/dx[f(x)g(x)] = f(x)•g’(x) + g(x)•f’(x)

‼️Lil trick is Left D Right + Right D Left

Or if its a polynomial equation then it’d be d/dx = u•v’ + v•u’

Question 2

How would you apply the product rule when differentiating this equation?
h(x)=2x(sin(x))

Answer 2

First of all I’d figure out what are the differentiable functions in this equation.

f(x) or u = 2x
g(x) or v = sin(x)

Then I’d find the derivative of these functions by utilizing the power rule.

f ‘(x) or u’ = 2
g ‘(x) or v’ = cos(x)

Now from this I’d use the equation of
f(x)g’(x)+g(x)f’(x)

Plugging in the values
2xcosx + sinx2

Which in result equal to
2xcos(x) + 2sin(x)

Question 3

What is the Quotient Rule of Differentiation?

Answer 3

First I would determine if the quotient rule is used, by first looking at what is given like a function above another.

f(x)/g(x) in this case

then when finding the derivative of that this would be

d/dx [f(x)/g(x)] = f ‘(x)g(x) - f(x)g ‘(x)/[g(x)]^2

Question 4

How would I apply the quotient rule when differentiating this equation?

h(x)=x^2/ln(x)

Answer 4

First, I’d determine if the differentiable function were to be in a fractional form.

In this case this is cleared as a check mark
h(x)=x^2/ln(x)

then I’d figure out which ones were f(x) / u and g(x) / v . After I’d find the derivative of each value

f(x) = x^2
f’(x) = 2x
g(x) = ln(x)
g’(x) = 1/x

After that step I plug those functions into the equation
h’(x) = (2x)(ln(x))-(x^2)(1/x)/|ln(x)|^2

This is final derivative
h’(x) = 2xlnx-x/|ln(x)|^2

Question 5

Explain the steps on how you would differentiate this equation

h(x)=x^2e^x sin (x)

Answer 5

First I would identify the functions given in this equation.

f(x)/u = x^2
g(x)/v = e^x
k(x)/q = sin(x)

Find the derivates of each one
f’(x)/u’ = 2x
g’(x)/v’ = e^x
k(x)/q’ = cos(x)

Then I would differentiate the product rule by following the formula:
h(x)=f(x)g(x)k(x) = sin(x)
DLMR+LDMR+LMDR

Then applying it to this format and plugging in the functions
h(x)=f’(x)g(x)k(x)+f(x)g’(x)k(x)+f(x)g(x)k’(x)

h’(x)=(2xe^xsin(x))+(x^2e^xsin(x))+(x^2e^xcos(x))

You get this as a result
2xe^xsin(x)+x^2e^xsin(x)+x^2e^xcos(x)

Question 6

Explain how you would differentiate this equation

h(x)=x^2+1/x*sin(x)

Answer 6

First I would understand what the functions are

First I would identify the functions in the equation

f(x) = x^2+1
f’(x) = 2x
g(x) = xsin(x)
(use the product rule for finding g’(x))
g’(x) = (xcos(x))+(sin(x)*1)= xcos(x) + sin(x)

Plug in values into the quotient rule equation
LoDHigh-HighDLo
(x sin(x) * 2x) - (x^2+1 * (xcos(x) + sin(x)))/(x*sin(x)^2