Differentiate the following function by applying the theorems. Write the solution of this answers

Question

1. f(x)=(2 - x³)(-10x + 5)

Answer: 40x³ -15x² -20

2. y= (x³-2x)/(x²+1)

Answer: x⁴ + 5x² -2/(x²+1)²

3. y = (5/x - 1)³

Answer: - 375/(x-1)⁴

4. If f(x)=4x² and g(x)=3x² +1

find (f o g)(x)

Answer: 48x(3x²+1)

Answer 1

Kevin, I will write each theorem and apply it.

1. f(x) = (2 - x³)( -10x + 5)

You can do this problem 2 ways. Multiply it out and use the power rule OR

Use the product rule and the chain rule.

f(x) = -20x + 10 + 10x⁴ - 5x² Combine like terms and write in depending order of powers

f(x) = 10 x⁴ - 5x³ - 20x + 10 Now apply the power rule to each term. f(x) = cxⁿ so f'(x) = cn x^n-1

f'(x) = 40x³ - 15x² -20

Now the same problem using the produce rule.

g(x)h(x) = g(x) h'(x) + g'(x) h(x)

for this problem g(x) = 2 - x³ and g'(x) = -3x² and h(x) = -10x+5 and h'(x) = -10

So. f'(x) = (2 - x³) (-10) + (-3x²) ( -10x+5)

f'(x) = -20 + 10x³ + 30x³ - 15x²

f'(x) = 40x³ - 15x² -20 Same answer. When I was in school we learned it as

"first times derivative of the second plus second times derivative of first".

Next problem:

y= (x³-2x)/(x²+1) This uses the quotient rule.

y = x³-2x y = hi y' = lo d(hi) less hi d(lo) over lo squared.

x²+1 lo

y' = (x²+1) ( 3x² - 2) - (x³-2x) ( 2x) Now multiply out the numerator and collect like terms.

(x²+1)² Leave the denominator as is.

Problem 3 is so much easier if you simplify it BEFORE you take the derivative.

y = (5/x - 1)³

y = [ 5 / ( x - 1 ) ]³ = 5³ /(x - 1)³ = 125 (x - 1) ^-3

Now take the derivative using the power rule.

y' = -3 (125) ( x - 1) ^-4

y' =. -375 / (x - 1)⁴

Problem 4 is a composite problem. It actually says "put g into f for all the x's"

So. f(x)=4x² and g(x)=3x² +1

(f o g) = 4 (3x² +1)²

(f o g)' = (2) (4) (3x² +1) (6x) The 6x comes from the chain rule. (The derivative of 3x² +1.)

(f o g)' = 48x (3x² +1)

I hope this helps.

Answer 2

desmos.com/calculator/wjyii4kaqd

The video addresses problem 2 of the post.