Question

Use the defintion of the derivative to find the derivative of g(x) =x3-2x2+x-1​

Answer 1

Answer:

The derivative is

3x²+4x

Step-by-step explanation:

The definition is:

f'(x)=limh→0f(x+h)−f(x)h

So the terms are:

f(x+h)=(x+h)3+2(x+h)2+1.

Term by term, we have:

(x+h)3=x3+3x2h+3xh2+h3

2(x+h)2=2(x2+2hx+h2)=2x2+4hx+2h2

Obviously,

f(x)=x3+2x2+1.

Now we compute f(x+h)−f(x), since it's very long, I'll just write down the result. If you'll need help with calculations don't hesitate to ask (but I strongly suggest to do them yourself, since it will improve your skill):

2h2+h3+4hx+3h2x+3hx2

Now we divide this quantity by

2

h

+

h

2

+

4

x

+

3

h

x

+

3

x

2

.

Now, taking the limit of this quantity as

h

→

0

means to erase all the terms involving

h

, leaving

3

x

2

+

4

x

as the final result