Q: Find the derivative of $$y = 4\sec(x)\tan(x).$$

$$4(\sec^3$(x) + \sec(x)\tan^2$(x))$$

Q: Find the derivative of $$y = 1 - \sec^2$(x)\sin(x).$$

$$-2\sec^2$(x)\sin(x)\tan(x) + \sec^2$(x)\cos(x)$$

Question 1

Evaluate the following limit: $$y = \lim_{x 	o 0} \sin(4x)$$

Accepted Answer

$$0$$

Question 2

Evaluate the following limit: $$y = \lim_{x 	o 0} \sin^2$(4x) + \cos^2$(3x)$$

Accepted Answer

$$1$$

Question 3

Find the limit: $$y = \lim_{x 	o 0} 	an(4x) \sin(3x)$$

Accepted Answer

$$0$$

Question 4

Find the derivative of $$y = 3x \sin(4x).$$

Accepted Answer

$$3\sin(4x) + 12x\cos(4x)$$

Question 5

Find the derivative of $$y = \cos^2$(2x) + \sin^3$(5x).$$

Accepted Answer

$$-4\sin(2x)\cos(2x) + 15\sin^2$(5x)\cos(5x)$$

Question 6

Find the derivative of $$y = 4x \cos(\sin(3x^2$)).$$

Accepted Answer

$$4\cos(\sin(3x^2$)) - 24x^2$ \sin(\sin(3x^2$)) \cos(3x^2$)$$

Question 7

Find the derivative of $$y = 4\sec(x)	an(x).$$

Accepted Answer

$$4(\sec^3$(x) + \sec(x)	an^2$(x))$$

Question 8

Find the derivative of $$y = 1 - \sec^2$(x)\sin(x).$$

Accepted Answer

$$-2\sec^2$(x)\sin(x)	an(x) + \sec^2$(x)\cos(x)$$

Question 9

Evaluate the following limit: $$y = \lim_{x 	o 0} \sin^3$(x) \cos(5x).$$

Accepted Answer

$$0$$

Question 10

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

Accepted Answer

$$2x\cot(x) - x^2$\csc^2$(x)$$

Question 11

A medical device uses the function $$y = 3x\sin(4x) to $$model the oscillation of a sensor. What is the instantaneous rate of change of the sensor's position at $$x = 0$$?

Accepted Answer

$$3\sin(0) + 12 \cdot 0 \cdot \cos(0) = 0$$

Question 12

A chemical reaction's rate is modeled by $$y = \cos^2$(2x) + \sin^3$(5x). $$What is the rate of change of the reaction at $$x = 0$$?

Accepted Answer

$$-4\sin(0)\cos(0) + 15\sin^2$(0)\cos(0) = 0$$

Question 13

Suppose $$y = x^2$ \cot(x)$$ models the angle of a robotic arm. What is the rate of change of the angle at $$x = \frac{\pi}{4}$$?

Accepted Answer

$$2 \cdot \frac{\pi}{4} \cot\left(\frac{\pi}{4}\right) - \left(\frac{\pi}{4}\$$right)^2$$ \csc^2$\left(\frac{\pi}{4}\right)$$

Question 14

A satellite's position is modeled by $$y = 4x \cos(\sin(3x^2$)). $$What is the derivative with respect to $$x$$?

Accepted Answer

$$4\cos(\sin(3x^2$)) - 24x^2$ \sin(\sin(3x^2$)) \cos(3x^2$)$$

Question 15

Which of the following best describes the behavior of $$\lim_{x 	o 0} \sin^3$(x) \cos(5x)$$?

Accepted Answer

The limit is $$0$$ because $$\sin^3$(x)$$ approaches $$0$$ faster than $$\cos(5x)$$ approaches any other value.

Step-by-Step Calculus Guidance: Limits and Derivatives

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