Q: A function is defined as $$y = \sqrt{\tan x}. $$What is $$\frac{dy}{dx}$$ for $$0 \u003c x \u003c \frac{\pi}{2}$$?

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

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}$$, where t$$ \u003e 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| \u003e 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 $$0 \u003c x \u003c \sqrt{\pi}$$?

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 $$0 \u003c x \u003c \frac{\pi}{2}.$$

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 $$0 \u003c x \u003c \frac{\pi}{2}$$?

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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