Math Core

This test covers the chain rule, implicit differentiation, derivatives of inverse functions and inverse trig functions, and higher-order derivatives.

Question 1

Find $\dfrac{dy}{dx}$ if $y = (5x^2 - 3)^4$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 2

Find $\dfrac{dy}{dx}$ if $y = e^{\sin(2x)}$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 3

The table gives values of differentiable functions $f$ and $g$ . If $h(x) = g(f(x))$ , what is $h'(2)$ ?

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
$-1$	0	5	2	4
$2$	$-1$	3	6	2

Question 4

Find $\dfrac{dy}{dx}$ if $y = \ln(\cos x)$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 5

Find $\dfrac{dy}{dx}$ if $x^3 + y^3 = 9$ . Give your answer in terms of $x$ and $y$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 6

The point $(1, 2)$ lies on the curve $x^2 + 3xy + y^2 = 11$ . Find the slope of the tangent line at that point.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 7

If $y^3 + y = x$ , then $\dfrac{dy}{dx} =$

Question 8

The point $(2, 1)$ lies on the curve $x^2y - y^2 = 3$ . Find an equation of the tangent line to the curve at that point.

Enter an expression, e.g. 3x^2 - 2x + 1

Question 9

Let $f(x) = x^3 + 4x + 2$ , and let $g$ be the inverse of $f$ . Find $g'(7)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 10

A differentiable, one-to-one function $f$ has $f(3) = 5$ , $f'(3) = 4$ , $f(5) = 9$ and $f'(5) = 2$ . If $g$ is the inverse of $f$ , what is $g'(5)$ ?

Question 11

Find $\dfrac{dy}{dx}$ if $y = \arctan(3x)$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 12

Let $f(x) = \arcsin(x^2)$ . Find $f'\!\left(\dfrac{1}{2}\right)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 13

Find $f''(x)$ if $f(x) = \cos(2x)$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 14

The point $(2, 1)$ lies on the curve $x^2 + 4y^2 = 8$ . Find the value of $\dfrac{d^2y}{dx^2}$ at $(2, 1)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 15

If $y = \sin(2x)$ , what is $\dfrac{d^{50}y}{dx^{50}}$ ?