Differentiation of polynomials...please help.?

Question

Can someone please help me find the the gradient of the curve y=x^3 - 5x^2 + 4 at the point where x=2

and...

If f(x)=2x^2 (x+1) then f ' (1) is equal to...

Answer 1

distribute

2x^3+2x^2

Differentiate both terms

6x^2+4x

replace 1 for x

6+4=10

Answer 2

1)

y = x³ - 5x² + 4

Now, when x = 2:

y = 2³ - 5(2²) + 4

y = 8 - 20 + 4

y = -8

Therefore, at point (2,-8)

Now, differentiate your equation:

y = x³ - 5x² + 4

dy/dx = 3x² - 10x

Now, plug your x value of 2 in to get your slope at that point:

3(2²) - 10(2)

Slope = -8

Using the point and the slope:

(y + 8) / (x - 2) = -8

-8(x - 2) = y + 8

-8x + 16 = y + 8

y = -8x + 8

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2)

f(x) = 2x²(x + 1)

We'll differentiate this using the product rule:

u = 2x²

u' = 4x

v = x + 1

v' = 1

f'(x) = u'v + uv'

f'(x) = 4x(x + 1) + 2x²(1)

f'(x) = 4x² + 4x + 2x²

f'(x) = 6x² + 4x

Therefore:

f'(1) = 6 + 4

f'(1) = 10