1.
Differentiate y= 3e^{-4x}
Correct Answer
C. Dy/dx= -12e^{-4x}
Explanation
The given function is y = 3e^(-4x). To differentiate this function with respect to x, we use the chain rule. The derivative of e^(-4x) is -4e^(-4x) and the derivative of 3 is 0. Therefore, the derivative of y with respect to x is dy/dx = 0 * e^(-4x) + 3 * (-4e^(-4x)) = -12e^(-4x).
2.
Is differentiation of e^{5x-1} is 5e^{5x-1} true or false?
Correct Answer
A. True
Explanation
The differentiation of e^5x-1 is 5e^5x-1. This can be derived using the chain rule of differentiation. The derivative of e^u is e^u times the derivative of u, which in this case is 5. Therefore, the correct answer is true.
3.
Differentiate y= 2e^{5x+2}
Correct Answer
B. Dy/dx= 10e^{5x+2}
Explanation
The correct answer is dy/dx= 10e5x+2. This is the correct answer because when differentiating y= 2e5x+2 with respect to x, the derivative of e5x is 5e5x and the derivative of 2 is 0. Therefore, the derivative of y with respect to x is 5e5x. Additionally, the constant term 2 remains unchanged. So the final answer is dy/dx= 10e5x+2.
4.
Differentiate y= e^{2-3x}
Correct Answer
D. Dy/dx= -3e^{2-3x}
Explanation
The correct answer is dy/dx= -3e^2-3x. To differentiate y = e^2-3x, we need to apply the chain rule. The derivative of e^2-3x with respect to x is -3e^2-3x.
5.
Is differentiation of e^{1+5x} is -5e^{1+5x} true or false?
Correct Answer
B. False
Explanation
The statement "Differentiation of e1+5x is -5e1+5x" is false. When differentiating e1+5x, the derivative of e1 is 0 since e1 is a constant. The derivative of 5x is simply 5. Therefore, the correct answer is false.
6.
Differentiate y= ln(7x+2)
Correct Answer
A. Dy/dx = 7 /(7x+2)
Explanation
The correct answer is dy/dx = 7 /(7x+2). This is the correct differentiation of the given function y= ln(7x+2). The derivative of ln(u) with respect to x is du/dx divided by u. In this case, u = 7x+2, so du/dx = 7. Therefore, the derivative of ln(7x+2) with respect to x is 7/(7x+2).
7.
Is Differentiation of ln(5x) is 1/x true or false?
Correct Answer
A. True
Explanation
The differentiation of ln(5x) can be found using the chain rule. The derivative of ln(u) is 1/u multiplied by the derivative of u with respect to x. In this case, u = 5x. So, the derivative of ln(5x) is 1/(5x) multiplied by the derivative of 5x, which is 5. Simplifying further, we get 1/x. Therefore, the statement "Differentiation of ln(5x) is 1/x" is true.
8.
Differentiate y =ln(x-8)
Correct Answer
B. Dy/dx = 1/(x-8)
Explanation
The derivative of ln(x-8) with respect to x is 1/(x-8). This can be found using the chain rule, where the derivative of ln(u) with respect to u is 1/u, and the derivative of (x-8) with respect to x is 1. Therefore, the derivative of ln(x-8) is 1/(x-8).
9.
Is differentiation of ln(4x^{2}-1) is 4x/(4x^{2}-1) true or false?
Correct Answer
B. False
Explanation
The differentiation of ln(4x^2-1) is not equal to 4x/(4x^2-1). The correct answer is false. To differentiate ln(4x^2-1), we need to apply the chain rule. The derivative of ln(u) is 1/u multiplied by the derivative of u with respect to x. Therefore, the correct differentiation of ln(4x^2-1) is (8x)/(4x^2-1).
10.
Differentiate y= ln(2+3x^{2})
Correct Answer
D. Dy/dx= 6x/(2+3x^{2})
Explanation
The given problem asks us to differentiate the function y = ln(2 + 3x^2). To do this, we can use the chain rule. Taking the derivative of the function with respect to x, we get dy/dx = (2(3x))/(2 + 3x^2). Simplifying further, we have dy/dx = 6x/(2 + 3x^2). So, the correct answer is dy/dx = 6x/(2 + 3x^2).