Differentiation

Question 1

Which of the following is the derivative of $y = x3 + 5?$

Not correct. Choice (a) is false.

Try again, the derivative of a constant is zero.

Not correct. Choice (b) is false.

Try again, remember $-d(xn) = nxn-1. dx$

Not correct. Choice (c) is false.

Try again, remember $d --(xn) = nxn-1 , dx$ you have integrated $3 y = x .$

Your answer is correct.

$d n n- 1 d 3 2 2 dx(kx ) = knx so dx(x + 5) = 3 ×x + 0 = 3x .$

Question 2

Which of the following is the derivative of $√-- f(x) = x ?$
Recall $n√x-= x 1n .$

Not correct. Choice (a) is false.

Try again, convert $f (x) = √x-= x12$ and then use the differentiation rules.

Not correct. Choice (b) is false.

Try again, you have integrated $√ -- x.$

Your answer is correct.

$√-- 12 ′ 1 - 12 -1-- --1- f(x) = x = x and f(x) = 2x = 2x12 = 2√x--.$

Not correct. Choice (d) is false.

Try again, convert $√-- 12 f (x) = x = x$ and then use the differentiation rules.

Question 3

Which of the following is the derivative of $1 y = x-?$

Not correct. Choice (a) is false.

Try again, convert $1- -1 y = x = x$ and then use the differentiation rules.

Not correct. Choice (b) is false.

Try again, look carefully at the sign.

Not correct. Choice (c) is false.

Try again, convert $y =-1= x- 1 x$ and then use the differentiation rules.

Your answer is correct.

$y = 1-= x-1 and dy-= - 1x-2 = --1 . x dx x2$

Question 4

Which of the following is the second derivative of $f(x) = - 2x4 + 3x-2 + 9?$

Not correct. Choice (a) is false.

Try again, check the signs

Not correct. Choice (b) is false.

Try again, this is the ﬁrst derivative.

Your answer is correct.

$f′(x) = - 8x3 - 6x-3, so f′′(x) = - 24x2 + 18x-4.$

Not correct. Choice (d) is false.

Try again, you must differentiate each term twice, noting that $f′(x) = - 8x3 - 6x- 3.$

Question 5

The curve $y = 2x3 - 9x2 + 12x- 3$ has a local minimum at which value of $x ?$

Your answer is correct.

$-dy dx$	=	$2 2 6x - 18x + 12 = 6(x - 3x +2) = 6(x - 1)(x - 2),$
$so -dy dx$	=	0 when $x = 1$ and $x = 2.$
$2 d-y dx2$	=	$12x - 18 .$
Hence $d2y --2 dx$	$<$	0 when $x = 1$ and $d2y --2 > 0 dx$ when $x = 2 .$

Hence there is a maximum at $x = 1$ and a minimum at $x = 2.$

Not correct. Choice (b) is false.

Try again, this is where the second derivative of the function is zero, not where the gradient is zero.

Not correct. Choice (c) is false.

Try again, 1 is the local maximum of the function.

Not correct. Choice (d) is false.

Try again, you may have used the wrong function, there is a minimum at $x = - 1$ for the function $y = 2x3 + 9x2 + 12x- 3.$


		Faculty of Economics and Business MathQuiz