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3.015 Higher derivatives

Worksheet
Higher derivatives
1

Suppose y = x^{3} + 24 x^{2} + 5 x + 65.

a

Find \dfrac{d y}{d x}.

b

Find \dfrac{d^{2} y}{d x^{2}}.

c

Find the value of x at which \dfrac{d^{2} y}{d x^{2}} = 0.

2

Suppose y = 5 x^{8}. Find:

a

\dfrac{d y}{d x}

b

\dfrac{d^{2} y}{d x^{2}}

c

\dfrac{d^{3} y}{d x^{3}}

d

\dfrac{d^{4} y}{d x^{4}}

e

\dfrac{d^{5} y}{d x^{5}}

3

Suppose y = \dfrac{2}{x}. Find:

a

\dfrac{d y}{d x}

b

\dfrac{d^{2} y}{d x^{2}}

c

\dfrac{d^{3} y}{d x^{3}}

d

\dfrac{d^{4} y}{d x^{4}}

4

Suppose y = \dfrac{2}{x^{5}}. Find:

a

\dfrac{d y}{d x}

b

\dfrac{d^{2} y}{d x^{2}}

c

\dfrac{d^{3} y}{d x^{3}}

5

Suppose y = \left( 3 x + 7\right)^{7}. Find:

a

\dfrac{d y}{d x}

b

\dfrac{d^{2} y}{d x^{2}}

c

\dfrac{d^{3} y}{d x^{3}}

6

Consider the function y = - 2 x^{4} + 4 x^{3} - 7 x^{2} + 8 x - 9. Find:

a

\dfrac{d y}{d x}

b

The value of \dfrac{d y}{d x} when x = 3

c

\dfrac{d^{2} y}{d x^{2}}

d

The value of \dfrac{d^{2} y}{d x^{2}} when x = 4

e

\dfrac{d^{3} y}{d x^{3}}

f

The value of \dfrac{d^{3} y}{d x^{3}} when x = 6

7

Find the second derivative of the following functions:

a
f \left( x \right) = 6 x^{4}
b
f \left( x \right) = 3 x^{2} + 7 x + 5
c
f \left( x \right) = \dfrac{8}{x^{2}}
d
f \left( x \right) = x^{\frac{1}{7}}
e
f \left( x \right) = \left( 4 x + 3\right)^{2}
f
f \left( x \right) = \left( 3 x + 2\right)^{4}
g
f \left( x \right) = \dfrac{4}{\left(x + 8\right)^{2}}
h
y = 4 \sqrt{6 - x}
i
y = \dfrac{9}{1 + x}
j
f \left( x \right) = \sqrt{x^{2} + 5}
8

Find the second derivative of the following functions:

a
y = \left(7 - 6 x^{2}\right)^{\frac{1}{3}}
b
f \left( x \right) = x^{2} \left(x^{3} + 3\right)
c
f \left( x \right) = \left(x^{4} + 2\right) \left(3 - 2 x^{3}\right)
9

For the function y = \left( 3 \cos x - 5\right)^{4}, find:

a

\dfrac{d y}{d x}

b

\dfrac{d^{2} y}{d x^{2}}

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Outcomes

4.1.1.1

understand the concept of the second derivative as the rate of change of the first derivative function

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