Evaluate [tex]$\lim _{x \rightarrow x}\left(1-\frac{3}{x}\right)^x$[/tex]
A. 3
B. [tex]$e^3$[/tex]
C. [tex]$e^{-3}$[/tex]
D. -3

If [tex]$x=t^2-1$[/tex] and [tex]$y=t^3+2$[/tex], find [tex]$\frac{d y}{d x}$[/tex].
A. [tex]$2 t$[/tex]
B. [tex]$3 t$[/tex]
C. [tex]$\frac{4}{3}$[/tex]
D. [tex]$\frac{28}{3}$[/tex]

If [tex]$y=(3 x+8)^{19}$[/tex], obtain [tex]$y'$[/tex]
A. [tex]$(3 x+8)^{18}$[/tex]
B. [tex]$27(3 x+8)^{18}$[/tex]
C. 5713
D. [tex]$19(3 x+8)^{12}$[/tex]

Find [tex]$\frac{d y}{d x}$[/tex] if [tex]$y=\cos ^2 x+\sin ^2 x$[/tex].
A. [tex]$-2 \sin x \cos x$[/tex]
B. 0
C. [tex]$2 \cos x+2 s$[/tex]

Find [tex]$\frac{d y}{d x}$[/tex] if [tex]$3+y^2=x^3-y$[/tex].
A. [tex]$\frac{x^2}{-2 x+1}$[/tex]
B. [tex]$\frac{3 x^3}{2 y-1}$[/tex]
C. [tex]$\frac{x^2}{2 x^2+1}$[/tex]
D. [tex]$\frac{3 r^2}{2 g+1}$[/tex]

Given [tex]$y(x)=3 x^2 r+\sin x$[/tex], find [tex]$y^{\prime \prime}(0)$[/tex]
A. 3
B. -12
C. 6
D. 12

Given [tex]$y=60 x^{-5 x}$[/tex], find [tex]$\frac{d y}{d x}$[/tex]
A. [tex]$6 e^{-5 x}(1-5 x)$[/tex]
B. [tex]$x e^{-i x}(5 x-1)$[/tex]
C. D. [tex]$6 e^{-4 x}(1+5 x)$[/tex]

Find the derivative of [tex]$y=\frac{x^2-1}{2 x+1}$[/tex] at [tex]$x=0$[/tex].
A. [tex]$\frac{1}{4}$[/tex]
B. [tex]$\frac{1}{3}$[/tex]
C. [tex]$\frac{1}{2}$[/tex]
D.

Find the derivative of [tex]$y=\ln \left(x^3\right)+\sin x$[/tex]
A. [tex]$\frac{3}{x}+\cos x$[/tex]
B. [tex]$\frac{1}{x}$[/tex]
C. [tex]$2 x+\cos x$[/tex]
D. [tex]$\sin x$[/tex]

Question 48 uses the limit definition to find the limit as x approaches infinity: e − 3 ​ .
Question 49 uses the chain rule to find dy/dx: 2 3 ​ t ​ .
Question 50 uses the chain rule to find the derivative: 57 ( 3 x + 8 ) 18 ​ .
Question 51 simplifies the trigonometric identity and finds the derivative: 0 ​ .
Question 52 uses implicit differentiation to find dy/dx: 2 y + 1 3 x 2 ​ ​ .
Question 53 calculates the second derivative and evaluates at x=0: 6 r ​ .
Question 54 calculates the derivative: − 30 e − 5 x ​ .
Question 55 uses the quotient rule to find the derivative at x=0: 2 ​ .
Question 56 finds the derivative using logarithm properties: x 3 ​ + cos x ​ .

Explanation

Introduction We are given a set of multiple-choice calculus problems. Let's solve each one step by step.

Question 48 Solution Question 48: Evaluate lim x → ∞ ​ ( 1 − x 3 ​ ) x . This is a standard limit of the form lim x → ∞ ​ ( 1 + x a ​ ) x = e a . Here, a = − 3 . Therefore, the limit is e − 3 .

Question 49 Solution Question 49: If x = t 2 − 1 and y = t 3 + 2 , find d x d y ​ . We use the chain rule: d x d y ​ = d t d y ​ / d t d x ​ . We have d t d x ​ = 2 t and d t d y ​ = 3 t 2 . Thus, d x d y ​ = 2 t 3 t 2 ​ = 2 3 ​ t .

Question 50 Solution Question 50: If y = ( 3 x + 8 ) 19 , find y ′ . We use the chain rule. Let u = 3 x + 8 . Then y = u 19 , so d u d y ​ = 19 u 18 and d x d u ​ = 3 . Thus, d x d y ​ = 19 ( 3 x + 8 ) 18 ".3 = 57 ( 3 x + 8 ) 18 .

Question 51 Solution Question 51: Find d x d y ​ if y = cos 2 x + sin 2 x . Since cos 2 x + sin 2 x = 1 , we have y = 1 . Therefore, d x d y ​ = 0 .

Question 52 Solution Question 52: Find d x d y ​ if 3 + y 2 = x 3 − y . We use implicit differentiation. Differentiating both sides with respect to x , we get 0 + 2 y d x d y ​ = 3 x 2 − d x d y ​ . Thus, ( 2 y + 1 ) d x d y ​ = 3 x 2 , so d x d y ​ = 2 y + 1 3 x 2 ​ .

Question 53 Solution Question 53: Given y ( x ) = 3 x 2 r + sin x , find y ′′ ( 0 ) . First, find the first derivative: y ′ ( x ) = 6 x r + cos x . Then find the second derivative: y ′′ ( x ) = 6 r − sin x . Evaluating at x = 0 , we get y ′′ ( 0 ) = 6 r − sin ( 0 ) = 6 r .

Question 54 Solution Question 54: Given y = 60 x − 5 x , find d x d y ​ . Actually, the question states y = 6 e − 5 x . Then d x d y ​ = 6 ⋅ ( − 5 ) e − 5 x = − 30 e − 5 x .

Question 55 Solution Question 55: Find the derivative of y = 2 x + 1 x 2 − 1 ​ at x = 0 . We use the quotient rule: d x d y ​ = ( 2 x + 1 ) 2 ( 2 x + 1 ) ( 2 x ) − ( x 2 − 1 ) ( 2 ) ​ = ( 2 x + 1 ) 2 4 x 2 + 2 x − 2 x 2 + 2 ​ = ( 2 x + 1 ) 2 2 x 2 + 2 x + 2 ​ . Evaluating at x = 0 , we get d x d y ​ ∣ x = 0 ​ = ( 2 ( 0 ) + 1 ) 2 2 ( 0 ) 2 + 2 ( 0 ) + 2 ​ = 1 2 ​ = 2 .

Question 56 Solution Question 56: Find the derivative of y = ln ( x 3 ) + sin x . Using logarithm properties, y = 3 ln x + sin x . Then d x d y ​ = x 3 ​ + cos x .

Final Answers Final Answers:

C. e − 3

B. 2 3 ​ t

C. 57 ( 3 x + 8 ) 18

B. 0

D. 2 y + 1 3 x 2 ​

C. 6 r

A. − 30 e − 5 x

C. 2

A. x 3 ​ + cos x


Examples
These calculus problems are fundamental in various fields. For instance, understanding limits (Question 48) is crucial in analyzing the behavior of algorithms and systems. Derivatives (Questions 49, 50, 51, 52, 53, 54, 55, 56) are essential in physics for calculating velocity and acceleration, in economics for marginal cost and revenue, and in computer graphics for creating smooth animations and realistic simulations. Mastering these concepts provides a strong foundation for advanced studies and practical applications.

Answered by GinnyAnswer | 2025-07-08

The answers to the multiple-choice problems are as follows: 1. C. e − 3 , 2. B. 2 3 ​ t , 3. C. 57 ( 3 x + 8 ) 18 , 4. B. 0, 5. B. 2 y + 1 3 x 2 ​ , 6. C. 6r, 7. A., 8. A. 4 1 ​ , 9. A. x 3 ​ + cos x .
;

Answered by Anonymous | 2025-07-13