Evaluate each of the following. Show all your calculations.
a) f'(���3)� if f(x) ���� = x^4�� − 3x�
b) f' ���(−2)� if ���f(x) = 2x^3�� + 4��x^2 − 5x� + 8
c) f" (����1)� if f(X) ���� = −3x^2�� − 5x� + 7
d)f" ����(−3�) if f(x) ���� = 4x^3�� − 3��x^2 + 2x� − 6
e) f'(���0)� if f(x) ���� = 14x��^2 + 3�x − 6
f)f" ����(4�) if ����f (x) = x^5 + x��^4 − x^3
g) ��� f" (1/3) �if f(x)���� = −2x^5�� + 2x� − 6
h) f�′ (3/4)�if ����f(x) = −3x^3�� − 7��x^2 + 4�x − 11
To evaluate each of the functions and find their derivatives, we will use the power rule and sum/difference rule of differentiation. The power rule states that if f(x) = x^n, then f'(x) = n * x^(n-1). The sum/difference rule states that if f(x) = g(x) +/- h(x), then f'(x) = g'(x) +/- h'(x).
a) To find f'(3) if f(x) = x^4 - 3x:
First, find the derivative of each term:
f'(x) = (4x^3) - (3)
Substitute x = 3 into the derivative formula:
f'(3) = (4(3^3)) - (3)
f'(3) = 108 - 3
f'(3) = 105
b) To find f'(-2) if f(x) = 2x^3 + 4x^2 - 5x + 8:
First, find the derivative of each term:
f'(x) = (6x^2) + (8x) - (5)
Substitute x = -2 into the derivative formula:
f'(-2) = (6(-2^2)) + (8(-2)) - (5)
f'(-2) = (6(4)) - (16) - (5)
f'(-2) = 24 - 16 - 5
f'(-2) = 3
c) To find f"(1) if f(x) = -3x^2 - 5x + 7:
First, find the derivative of each term:
f'(x) = (-6x) - 5
Then, find the derivative of f'(x):
f"(x) = -6
Substitute x = 1 into the second derivative formula:
f"(1) = -6
d) To find f"(-3) if f(x) = 4x^3 - 3x^2 + 2x - 6:
First, find the derivative of each term:
f'(x) = (12x^2) - (6x) + 2
Then, find the derivative of f'(x):
f"(x) = (24x) - 6
Substitute x = -3 into the second derivative formula:
f"(-3) = (24(-3)) - 6
f"(-3) = -72 - 6
f"(-3) = -78
e) To find f'(0) if f(x) = 14x^2 + 3x - 6:
First, find the derivative of each term:
f'(x) = (28x) + 3
Substitute x = 0 into the derivative formula:
f'(0) = (28(0)) + 3
f'(0) = 0 + 3
f'(0) = 3
f) To find f"(4) if f(x) = x^5 + x^4 - x^3:
First, find the derivative of each term:
f'(x) = (5x^4) + (4x^3) - (3x^2)
Then, find the derivative of f'(x):
f"(x) = (20x^3) + (12x^2) - (6x)
Substitute x = 4 into the second derivative formula:
f"(4) = (20(4^3)) + (12(4^2)) - (6(4))
f"(4) = (20(64)) + (12(16)) - (6(4))
f"(4) = 1280 + 192 - 24
f"(4) = 1448
g) To find f"(1/3) if f(x) = -2x^5 + 2x - 6:
First, find the derivative of each term:
f'(x) = (-10x^4) + 2
Then, find the derivative of f'(x):
f"(x) = (-40x^3)
Substitute x = 1/3 into the second derivative formula:
f"(1/3) = (-40(1/3)^3)
f"(1/3) = (-40/27)
h) To find f'(3/4) if f(x) = -3x^3 - 7x^2 + 4x - 11:
First, find the derivative of each term:
f'(x) = (-9x^2) - (14x) + 4
Substitute x = 3/4 into the derivative formula:
f'(3/4) = (-9(3/4)^2) - (14(3/4)) + 4
f'(3/4) = (-9(9/16)) - (14(3/4)) + 4
f'(3/4) = (-81/16) - (42/4) + 4
f'(3/4) = (-81/16) - (42/4) + 64/16
f'(3/4) = (-81/16) - (168/16) + 64/16
f'(3/4) = (-249/16) + (64/16)
f'(3/4) = -185/16