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# Calculate the derivative of the following function: f (x

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MATHEMATICS - Sample Test 2 - with solutions 1. Calculate the area of the trangle with vertices A(1, 1, 0), B(0, 1, 0), C(0, 0, 1).

2. Calculate the derivative of the following function:

f (x) = e2x x2 3. Calculate the derivative of the following function:

f (x) =p

sin(ln(2x)) SOLUTIONS 1. Find the vectors:

AB = [−1, 0, 0] ~~ AC = [−1, −1, 1].

Their cross product:

AB × ~~ AC =

~i ~j ~k

−1 0 0

−1 −1 1

= 0 ·~i + 1 · ~j + 1 · ~k = [0, 1, 1].

The length of the obtained vector:

| ~AB × ~AC| =√

02+ 12+ 12 =√ 2.

The area of the triangle ABC:

Area = 1

2| ~AB × ~AC| =√

02+ 12+ 12 =

√2 2 . 2. Use the quotient rule

f0(x) = e2x x2

0

= (e2x)0· x2− e2x· (x2)0

(x2)2 = 2e2x· x2− e2x· 2x

x4 .

3. Use three times the chain rule f0(x) = (p

sin(ln(2x)))0 = 1

2psin(ln(2x)) · (sin(ln(2x)))0 =

= 1

2psin(ln(2x)) · cos(ln(2x)) · (ln(2x))0 =

= 1

2psin(ln(2x)) · cos(ln(2x)) 1

2x· (2x)0 = 1

2psin(ln(2x)) · cos(ln(2x)) 1 2x · 2.

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