Differentiation Basic Rules Quiz

Explanation
The correct answer is 12x3-5. This is the correct answer because when we differentiate the given function y=3x4-5x, we apply the power rule of differentiation which states that when differentiating a term with a variable raised to a power, we multiply the coefficient by the power and then decrease the power by 1. Therefore, when we differentiate 3x4, we get 12x3, and when we differentiate -5x, we get -5. Thus, the derivative of y=3x4-5x is 12x3-5.

Explanation
The differentiation of y=6x^3-1 is not equal to 18x^3. When we differentiate y=6x^3-1, the power rule states that the derivative of x^n is nx^(n-1). Therefore, the derivative of 6x^3 is 18x^2, not 18x^3.

Explanation
The given expression is y = 5/x^2. To differentiate this expression, we use the power rule of differentiation. The power rule states that when differentiating a function of the form f(x) = x^n, the derivative is given by f'(x) = nx^(n-1). Applying this rule, we differentiate y = 5/x^2 as follows:
y' = (5)(-2)x^(-2-1) = -10x^(-3) = -10/x^3. Therefore, the correct answer is -10x-3.

Explanation
The given expression is a polynomial with two terms. To differentiate it, we can use the power rule, which states that the derivative of x^n is n*x^(n-1). Applying this rule to the first term, -3x, we get -3. For the second term, -10x, the derivative is -10. Therefore, the derivative of the given expression is 15x - 3 - 10, which simplifies to 15x - 6 - 10.

Explanation
The given expression is y=3x-4+6x-1. We can simplify this expression by combining like terms. Combining 3x and 6x, we get 9x. Combining -4 and -1, we get -5. Therefore, the simplified expression is 9x - 5. The answer -12x-5+6 does not match the simplified expression, so it is not the correct answer.

Explanation
The given expression is y=5-13x^2-4x^3. To differentiate this expression, we take the derivative of each term. The derivative of 5 is 0, the derivative of -13x^2 is -26x, and the derivative of -4x^3 is -12x^2. Therefore, the differentiated expression is -26x-12x^2.

Explanation
The given expression y=5x-2-12 can be simplified as y=5x-14. Therefore, the correct answer -10x-3 is false.

Explanation
The given expression is y = 4/x^4 - 4x - 5. To differentiate this expression, we can use the power rule for differentiation. The derivative of 4/x^4 is -16/x^5, the derivative of -4x is -4, and the derivative of -5 is 0. Combining these derivatives, we get -16/x^5 - 4. Simplifying further, we can rewrite this as -16x^-5 - 4, which is equivalent to -16/x^5 - 4. Therefore, the correct answer is -16x - 5.