Q: A function is defined as $y = \ln(\sin(x^2))$. What is $\frac{dy}{dx}$ for $0 < x < \sqrt{\pi}$?

$\frac{2x\cos(x^2)}{\sin(x^2)}$

Q: Let $y = \tan(\sqrt{x})$. What is $\frac{dy}{dx}$?

$\frac{\sec^2(\sqrt{x})}{2\sqrt{x}}$

Q: A function is defined as $y = \sqrt{\tan x}$. What is $\frac{dy}{dx}$ for $0 < x < \frac{\pi}{2}$?

$\frac{\sec^2 x}{2\sqrt{\tan x}}$

Question 1

Suppose a medication's concentration in the bloodstream is modeled by $C(t) = e^{0.2t^2}$, where $t$ is time in hours. What is the rate of change of concentration with respect to time at $t=1$ hour?

Accepted Answer

$0.4e^{0.2}$

Question 2

Let $y = \ln(x^2 + 5)$. What is $\frac{dy}{dx}$?

Accepted Answer

$\frac{2x}{x^2 + 5}$

Question 3

A population of bacteria grows according to $P(t) = 3^{t^2 \log_3 t}$, where $t > 0$. What is $\frac{dP}{dt}$?

Accepted Answer

$3^{t^2 \log_3 t} \left(2t \log_3 t + \frac{t^2}{t \ln 3}\right)$

Question 4

Find the derivative of $y = 	an^{-1}(x^2)$ with respect to $x$.

Accepted Answer

$\frac{2x}{1 + x^4}$

Question 5

If $y = \sec^2(2x)$, what is $\frac{dy}{dx}$?

Accepted Answer

$4\sec^2(2x)	an(2x)$

Question 6

Let $y = \sin^2(3x - 2)$. What is $\frac{dy}{dx}$?

Accepted Answer

$6\sin(3x - 2)\cos(3x - 2)$

Question 7

A chemical reaction's rate is given by $R(t) = \ln(5e^{2t} + 1)$. What is $\frac{dR}{dt}$?

Accepted Answer

$\frac{10e^{2t}}{5e^{2t} + 1}$

Question 8

Find the derivative of $y = e^{\cot x}$.

Accepted Answer

$-e^{\cot x}\csc^2 x$

Question 9

Let $y = \sec^{-1}(x^2)$. What is $\frac{dy}{dx}$ for $|x| > 1$?

Accepted Answer

$\frac{2}{|x|\sqrt{x^4 - 1}}$

Question 10

If $y = (4 + 5x)^6$, what is $\frac{dy}{dx}$?

Accepted Answer

$30(4 + 5x)^5$

Question 11

A function is defined as $y = \ln(\sin(x^2))$. What is $\frac{dy}{dx}$ for $0 < x < \sqrt{\pi}$?

Accepted Answer

$\frac{2x\cos(x^2)}{\sin(x^2)}$

Question 12

Find the derivative of $y = x^2 \ln(x)$.

Accepted Answer

$2x\ln(x) + x$

Question 13

Let $y = 	an(\sqrt{x})$. What is $\frac{dy}{dx}$?

Accepted Answer

$\frac{\sec^2(\sqrt{x})}{2\sqrt{x}}$

Question 14

A function is given by $y = \sin(\ln x)$. What is $\frac{dy}{dx}$?

Accepted Answer

$\frac{\cos(\ln x)}{x}$

Question 15

Find the derivative of $y = \ln(	an x)$ for $0 < x < \frac{\pi}{2}$.

Accepted Answer

$\sec^2 x / 	an x$

Question 16

Let $y = e^{x^2 + 3x}$. What is $\frac{dy}{dx}$?

Accepted Answer

$e^{x^2 + 3x}(2x + 3)$

Question 17

A function is defined as $y = \sqrt{	an x}$. What is $\frac{dy}{dx}$ for $0 < x < \frac{\pi}{2}$?

Accepted Answer

$\frac{\sec^2 x}{2\sqrt{	an x}}$

Question 18

Find the derivative of $y = \ln(\cos x)$.

Accepted Answer

$-	an x$

Question 19

Let $y = \sin(\sin(2x))$. What is $\frac{dy}{dx}$?

Accepted Answer

$2\cos(2x)\cos(\sin(2x))$

The Chain Rule; Derivatives of Exponential, Logarithmic & Inverse Trigonometric Functions

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