1.
Which method should we use for y=(x-2)^{3}(x+1)?
Correct Answer
B. Product Rule
Explanation
The given function y=(x-2)3(x+1) involves the multiplication of two functions, (x-2) and (x+1). Therefore, the appropriate method to differentiate this function is the Product Rule. The Product Rule states that when differentiating the product of two functions, u and v, the derivative is given by the first function times the derivative of the second function, plus the second function times the derivative of the first function. In this case, u = (x-2) and v = (x+1), so applying the Product Rule will yield the correct result.
2.
Which method should use for y = x^{5}/(x-3)?
Correct Answer
C. Quotient Rule
Explanation
The given function is in the form of a quotient, where the numerator is x^5 and the denominator is (x-3). The Quotient Rule is used to differentiate such functions. According to the Quotient Rule, the derivative of a quotient is calculated by taking the derivative of the numerator (5x^4) multiplied by the denominator (x-3), minus the derivative of the denominator (1), multiplied by the numerator. Therefore, the Quotient Rule should be used to find the derivative of y = x^5/(x-3).
3.
The differentiation of 7x^{-3} is 21x^{-4}.
Correct Answer
B. False
4.
Differentiate y= -2x^{5 }- 6x^{2} + 3.
Correct Answer
D. -10x^{4}-12x
Explanation
The given answer is obtained by differentiating the given function, y = -2x^5 - 6x^2 + 3, using the power rule of differentiation. According to this rule, when differentiating a term of the form ax^n, the derivative is obtained by multiplying the coefficient a by the exponent n and then subtracting 1 from the exponent. Applying this rule to each term in the function, we get -10x^4 - 12x. Therefore, the correct answer is -10x^4 - 12x.
5.
Y=x^{4}(x-2)^{4} can be solved by Quotient Rule. True or false
Correct Answer
B. False
Explanation
The given expression y=x^4(x-2)^4 does not require the Quotient Rule to be solved. The Quotient Rule is used to differentiate functions that involve a quotient of two functions. In this case, there is no division involved, so the Quotient Rule is not applicable. Therefore, the correct answer is false.
6.
Y= (x+4)/(x-3)^{4} can be solved by Quotient Rule. True or false
Correct Answer
A. True
Explanation
The given equation y= (x+4)/(x-3) can be solved using the Quotient Rule, which is a method used to find the derivative of a function that is in the form of a quotient. The Quotient Rule states that if y= f(x)/g(x), then the derivative of y with respect to x is given by (g(x)f'(x) - f(x)g'(x))/(g(x))^2. Since the equation y= (x+4)/(x-3) is in the form of a quotient, it can be solved using the Quotient Rule. Therefore, the statement "y= (x+4)/(x-3) can be solved by Quotient Rule" is true.
7.
Differentiate y= 6x - sin (3x).
Correct Answer
B. 6 - 3 cos (3x)
Explanation
The correct answer is 6 - 3 cos (3x). To differentiate the given function, we first differentiate the term 6x, which gives us 6. Then, we differentiate the term -sin(3x) using the chain rule, which gives us -3 cos(3x). Therefore, the derivative of y = 6x - sin(3x) is 6 - 3 cos(3x).
8.
Differentiate y = 4x^{3} - e^{2x}.
Correct Answer
C. 12x^{2 }- 2e^{2x}
9.
Differentiate y = x(x^{2}-6).
Correct Answer
B. 3x^{2 }- 6
Explanation
The correct answer is 3x^2 - 6. This is the correct differentiation of the given function y = x(x^2-6). To differentiate this function, we use the power rule and the product rule. The power rule states that the derivative of x^n is n*x^(n-1), and the product rule states that the derivative of the product of two functions is the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function. Applying these rules, we get the derivative of y = x(x^2-6) as 1*(x^2-6) + x*(2x) = x^2 - 6 + 2x^2 = 3x^2 - 6.