Greetings,

So I've been working on this problem:
H(x) = (x^4 - 2x +7)(x^-3 + 2x^-4)
H'(x)=
Kept on getting it wrong and I assumed it was an algebra mistake. After multiple tries I went to a few derivative calculators to check my work...
What I saw was the calculators converted the problem to:

((x+2)*(x^4 -2x +7))/x^4

How exactly do I get from the first equation to the second?

When there is a negavtive exponent you need to flip the term to makee it positive so the problem should be written as:

H(x)=(x^4-2x+7)[(1/x^3)+(2/x^4)]

Then you simply multiply the two groups to get:
H(x)=[(x^4-2x+7)/(x^3)]+[2(x^4-2x+7)/(x^4)]

Then simplify the second fraction to get:
H(x)=[(x^4-2x+7)/(x^3)] + (2x^4/x^4) - (4x/x^4) + (14/x^4)
H(x)=[(x^4-2x+7)/(x^3)] + 2 - (4/x^3) + (14/x^4)

Then sort by order of highest exponent to lowest and your final answer is:
H(x)=(14/x^4) -(4/x^3) +[(x^4-2x+7)/(x^3)] + 2

Whew

[(1/x^3)+(2/x^4)] = x/x^4 + 2/x^4 = (x+2)/x^4

Hello there! I'll be happy to explain the process of simplifying the expression from your original equation H(x) to the form you saw in the derivative calculators.

Let's break down the simplification step by step:

Step 1: Expand the product
H(x) = (x^4 - 2x + 7)(x^-3 + 2x^-4)
= x^4 * x^-3 + x^4 * 2x^-4 - 2x * x^-3 - 2x * 2x^-4 + 7 * x^-3 + 7 * 2x^-4
= x^1 - x^-1 + 2 - 4x^-1 + 7x^-3 + 14x^-4

Step 2: Combine like terms
Now, let's rewrite the expression by combining like terms:
H(x) = x - x^-1 + 2 - 4x^-1 + 7x^-3 + 14x^-4

Step 3: Factor out common factors
Observe that x^-1 is a common factor in the second and fourth terms. Similarly, x^-3 is a common factor in the fifth term, and x^-4 is a common factor in the last term. We can factor them out as follows:

H(x) = x - x^-1 + 2 - 4x^-1 + 7x^-3 + 14x^-4
= x - 4x^-1 + 2 + 7x^-3 + 14x^-4

Step 4: Simplify the first two terms
Next, let's simplify the second term by combining x^-1 terms:
H(x) = x - 4x^-1 + 2 + 7x^-3 + 14x^-4
= x - 4x^-1 + 2 + 7x^-3 + 14x^-4
= (1 - 4x^-1) + 2 + 7x^-3 + 14x^-4
= (1 - 4/x) + 2 + 7x^-3 + 14x^-4
= (1 - 4/x) + 2 + 7/x^3 + 14/x^4

Step 5: Combine the simplified terms
Finally, combine all the terms to arrive at the simplified expression:
H(x) = (1 - 4/x) + 2 + 7/x^3 + 14/x^4
= (1 - 4/x) + 2 + 7/x^3 + 14/x^4
= (2 + 1) - 4/x + 7/x^3 + 14/x^4
= 3 - 4/x + 7/x^3 + 14/x^4
= (3x^4 - 4 + 7x + 14x^2) / x^4

Therefore, the second form you saw in the derivative calculators is:

H(x) = (3x^4 - 4 + 7x + 14x^2) / x^4

I hope this explanation helps you understand the process of simplifying the expression! Let me know if you have any further questions.