Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# Find the derivative of the function using the definition of derivative. f(x) = 2x^4?

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• 1 month ago

f(x + h) = 2(x + h)^4 = 2(x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4)

= 2x^4 + 8x^3h + 12x^2h^2 + 8xh^3 + 2h^4

f(x + h) - f(x) = f(x + h) = 2x^4 = 8x^3h + 12x^2h^2 + 8xh^3 + 2h^4

[f(x + h) - f(x)] / h = 8x^3 + 12x^2h + 8xh^2 + 2h^3

f'(x) = lim h-->0 [f(x + h) - f(x)] / h = 8x^3

All the other terms have a factor of h and vanish as h-->0.

• 1 month ago

2x^4

bring 4 to the front so:

4(2)x^

is 8x^ now:

now the exponent you subtract one from all exponents:

4-1=3 so

8x^3

• 1 month ago

f(x) = 2x^4

f'(x) = 4(2)x^(4-1)

f'(x) = 8x^3

• 1 month ago

The definition of a derivative is:

. . . . . f(x + h) - f(x)

lim . . -----------------

h->0 . . . . . h

You have:

f(x) = 2x^4

You can also figure out that:

f(x + h) = 2(x + h)^4

The hard part is to now expand that binomial raised to the 4th power. I simply used the expansion for (a + b)^n = C(n,0) a^n b^0 + C(n,1) a^(n-1) b^1 + ... + C(n,n) a^0 b^n.

Or you can expand it all out by distributing.

f(x + h) = 2(x + h)(x + h)(x + h)(x + h)

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

f(x + h) = 2[x²(x² + 2hx + h²) + 2hx(x² + 2hx + h²) + h²(x² + 2hx + h²)]

etc.

However you do it, you'll get to:

f(x + h) = 2[x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4]

f(x + h) = 2x^4 + 8x^3h + 12x^2h^2 + 8xh^3 + 2h^4

When you subtract f(x), the first term disappears:

f(x + h) - f(x) = 8x^3h + 12x^2h^2 + 8xh^3 + 2h^4

Next divide it by h:

[f(x + h) - f(x)]/h = 8x^3 + 12x^2h + 8xh^2 + 2h^3

When you take the limit as h approaches 0, the last 3 terms disappear.

8x^3

• 1 month ago

f '(x)=

limit {2[(x+h)^4-x^4]/h}, where h is a small increment in x.

h->0

=

limit {2[(x+h)^2+x^2][(x+h)^2-x^2]/h}

h->0

=

limit {2[(x+h)^2+x^2][2hx+h^2]/h}

h->0

=

2[x^2+x^2][2x]

=

8x3

• 1 month ago

f(x) = 2 x^4

d/dx(2 x^4) = 8 x^3

• rotchm
Lv 7
1 month ago

Recall that the definition is [f(x + h) - f(x)] / h

as h approaches zero; it's a limit.

So, what if f(x) ?

What is f(x+h).

Subtracting these & simplifying gives your numerator.

Divide it all now by h & simplify.

What do you get?

Then evaluate this as h gets very small.