49. If [tex]$x=x^4+1$[/tex] and [tex]$y=x^3+2$[/tex], find [tex]$\frac{d y}{d x}$[/tex].
A. 21
B. [tex]$3 t$[/tex]
C. [tex]$\frac{3}{2}$[/tex]
D. [tex]$\frac{34}{3}$[/tex]
50. [tex]$11 y=(3 x+8)^{19}$[/tex], obtain [tex]$y^{\prime}$[/tex]
A. [tex]$(3 a r+8)^{13}$[/tex]
B. [tex]$27(3 x+8)^{18}$[/tex]
C. [tex]$57(3 x+8)$[/tex] D. [tex]$19(3,2+8)^{18}$[/tex]
51. Find [tex]$\frac{d g}{d x}$[/tex] if [tex]$y=\cos ^2 x+\sin ^2 x$[/tex]. A. [tex]$-2 \sin x \cos x$[/tex] B. 0
C. [tex]$2 \cos x+2 \sin x$[/tex] D.
52. Find [tex]$\frac{4 x}{4 x}$[/tex] it [tex]$4+y^2=x^4 \quad 4$[/tex].
A. [tex]$\frac{3 c^2}{-4}$[/tex],
B. [tex]$\frac{3 x^2}{2 y-1}$[/tex]
C. [tex]$\frac{3 x^2}{2 y^2+1}$[/tex]
D. [tex]$\frac{3 x^2}{2 y+1}$[/tex]
53. Given [tex]$u(x)=3 x^2+\sin x$[/tex], find [tex]$y^{\prime \prime}(\theta)$[/tex]. A. 3 B , - 12 C. 6 D. 12
54. Given [tex]$y=6 x^{-5 x}$[/tex]. find [tex]$\frac{4}{4 x}$[/tex]. A. [tex]$6 x^{-5 x}(1-5 x)$[/tex]
B. [tex]$x e^{-5 x}(5 x$[/tex]
1)
C. [tex]$5^{5 x}$[/tex] D. [tex]$6 e^{-k x}(1+5 x)$[/tex]
55. 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. 2
56. Find the derivative of [tex]$y=\ln \left(x^3\right)+\sin x$[/tex]
A. [tex]$\frac{3}{x}+\cos x$[/tex]
B. [tex]$\frac{1}{x}+\cos x$[/tex] [tex]$2 x+\cos x$[/tex]
D. [tex]$\sin x$[/tex]

Question 49: d x d y ​ = 3 x 2
Question 50: y ′ = 11 57 ​ ( 3 x + 8 ) 18
Question 51: d x d y ​ = 0
Question 52: d x d y ​ = y 2 x 3 ​
Question 53: u ′′ ( x ) = 6 − sin x
Question 54: d x d y ​ = − 30 x − 5 x ( ln x + 1 )
Question 55: y ′ ( 0 ) = 2
Question 56: $y' = \frac{3}{x} + \cos x

Explanation

Problem Overview We are given a set of differential calculus problems to solve, ranging from finding derivatives of various functions to implicit differentiation. We will address each question individually, showing all steps clearly.

Question 49 Solution Question 49: Given x = x 4 + 1 and y = x 3 + 2 , find d x d y ​ .


First, we differentiate y with respect to x :
d x d y ​ = d x d ​ ( x 3 + 2 ) = 3 x 2 Since the options are numerical, we need to find the value of x . However, the equation x = x 4 + 1 is difficult to solve analytically. The problem is likely misstated or incomplete. Assuming the question meant to ask for d x d y ​ in terms of x , the answer is 3 x 2 . Without a specific value for x , we cannot choose a numerical answer. If we assume x = 1 , then d x d y ​ = 3 ( 1 ) 2 = 3 . However, x = 1 does not satisfy x = x 4 + 1 since 1  = 1 4 + 1 = 2 . Since we cannot determine the value of x , we cannot determine a numerical answer. The closest answer choice is 3 x , but it should be 3 x 2 . Thus, we cannot determine the correct answer.

Question 50 Solution Question 50: Given 11 y = ( 3 x + 8 ) 19 , find y ′ .

We differentiate both sides with respect to x :
11 d x d y ​ = 19 ( 3 x + 8 ) 18 ⋅ 3 d x d y ​ = 11 57 ​ ( 3 x + 8 ) 18 Thus, y ′ = 11 57 ​ ( 3 x + 8 ) 18 . None of the answer choices match this result.

Question 51 Solution Question 51: Find d x d y ​ if y = cos 2 x + sin 2 x .

We know that cos 2 x + sin 2 x = 1 , so y = 1 . Therefore, d x d y ​ = d x d ​ ( 1 ) = 0 Thus, the answer is 0.

Question 52 Solution Question 52: Find d x d y ​ if 4 + y 2 = x 4 + 4 .

Simplifying, we have y 2 = x 4 . Differentiating both sides with respect to x , we get: 2 y d x d y ​ = 4 x 3 d x d y ​ = 2 y 4 x 3 ​ = y 2 x 3 ​ None of the answer choices match this result.

Question 53 Solution Question 53: Given u ( x ) = 3 x 2 + sin x , find u ′′ ( x ) .

First, we find the first derivative: u ′ ( x ) = 6 x + cos x Then, we find the second derivative: u ′′ ( x ) = 6 − sin x Since the question asks for y ′′ ( θ ) , it's likely a typo and should be u ′′ ( x ) . None of the answer choices match this result.

Question 54 Solution Question 54: Given y = 6 x − 5 x , find d x d y ​ .

Taking the natural logarithm of both sides: ln y = ln ( 6 x − 5 x ) = ln 6 − 5 x ln x Differentiating both sides with respect to x :
y 1 ​ d x d y ​ = − 5 ( ln x + x ⋅ x 1 ​ ) = − 5 ( ln x + 1 ) d x d y ​ = y ( − 5 ) ( ln x + 1 ) = 6 x − 5 x ( − 5 ) ( ln x + 1 ) = − 30 x − 5 x ( ln x + 1 ) None of the answer choices match this result.

Question 55 Solution Question 55: Find the derivative of y = 2 x + 1 x 2 − 1 ​ at x = 0 .

Using the quotient rule: y ′ = ( 2 x + 1 ) 2 ( 2 x ) ( 2 x + 1 ) − ( 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 :
y ′ ( 0 ) = ( 2 ( 0 ) + 1 ) 2 2 ( 0 ) 2 + 2 ( 0 ) + 2 ​ = 1 2 ​ = 2 Thus, the answer is 2.

Question 56 Solution Question 56: Find the derivative of y = ln ( x 3 ) + sin x .

First, rewrite y = 3 ln x + sin x . Then, y ′ = x 3 ​ + cos x Thus, the answer is x 3 ​ + cos x .
Examples
Differential calculus is used extensively in physics to describe motion, such as calculating velocity and acceleration. For example, if you have a function that describes the position of a car over time, you can use derivatives to find the car's velocity and acceleration at any given moment. This is crucial for designing safer and more efficient vehicles.

Answered by GinnyAnswer | 2025-07-08