Question

Suppose that ƒ is a function given as f(x) = 4x² + 5x + 3.

Simplify the expression f(x + h).
f(x + h)
Simplify the difference quotient,
ƒ(x + h) − ƒ(x)
h
=
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The derivative of the function at x is the limit of the difference quotient as h approaches zero.
f(x+h)-f(x)
f'(x) =lim
h→0
h
ƒ(x + h) − f(x)
h
=

Answer 1

f(x +h) = 4x² +4h² +8xh +5x +5h +3

(f(x+h) -f(x))/h = 4h +8x +5

f'(x) = 8x +5

Explanation:

For f(x) = 4x² +5x +3, you want the simplified expression f(x+h), the difference quotient (f(x+h) -f(x))/h, and the value of that at h=0.

F(x+h)

Put (x+h) where h is in the function, and simplify:

f(x+h) = 4(x+h)² +5(x+h) +3

= 4(x² +2xh +h²) +5x +5h +3

f(x +h) = 4x² +4h² +8xh +5x +5h +3

Difference quotient

The difference quotient is ...

(f(x+h) -f(x))/h = ((4x² +4h² +8xh +5x +5h +3) - (4x² +5x +3))/h

= (4h² +8xh +5h)/h

(f(x+h) -f(x))/h = 4h +8x +5

Limit

When h=0, the value of this is ...

f'(x) = 4·0 +8x +5

f'(x) = 8x +5

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Additional comment

Technically, the difference quotient is undefined at h=0, because h is in the denominator, and we cannot divide by 0. The limit as h→0 will be the value of the simplified rational expression that has h canceled from every term of the difference. This will always be the case for difference quotients for polynomial functions.

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