Advanced Differentiation Assessment Test

Explanation
The derivative of (y + 5) / (y2 – 1) with respect to y is (-y2 – 10y – 1) / (y2 – 1)2.

Explanation
The given function is y = 52x-1. To find dy/dx, we need to take the derivative of y with respect to x. The derivative of 52x-1 is 52 * (-1) * x^(1-1) = -52x^0 = -52. Taking the derivative of ln 25 with respect to x gives us 0, as ln 25 is a constant. Therefore, the derivative of y = 52x-1 ln 25 is -52.

Explanation
The given expression is ax^2 + c. To differentiate this expression with respect to x and raise it to the 1/2 power, we can use the power rule of differentiation. The power rule states that when differentiating a function of the form x^n, the result is nx^(n-1). Applying this rule, we differentiate ax^2 to get 2ax, and since we are raising it to the 1/2 power, the final answer is 2ax.

Explanation
The given correct answer is -4 / [(4x)^2 – 1]^0.5. To find the derivative of arccos 4x with respect to x, we can use the chain rule. The derivative of arccos u with respect to u is -1 / (1 - u^2)^0.5, where u = 4x. Therefore, the derivative of arccos 4x with respect to x is -4 / [(4x)^2 – 1]^0.5.

Explanation
The given expression is (y + 1)3 - y3. To find its derivative with respect to y, we can apply the power rule of differentiation. The power rule states that if we have a term of the form (ax)n, its derivative is n(ax)^(n-1). Applying this rule to the first term, we get the derivative of (y + 1)3 as 3(y + 1)2. Similarly, the derivative of y3 is 3y^2. Since the two terms have opposite signs, their derivatives will also have opposite signs. Therefore, the derivative of (y + 1)3 - y3 with respect to y is 3(y + 1)2 - 3y^2, which simplifies to 6y + 3.

Explanation
The derivative of y = za can be found using the power rule of differentiation. According to the power rule, the derivative of x^n is n*x^(n-1). In this case, we can rewrite y = za as y = z^a. Applying the power rule, the derivative of y with respect to z is a*z^(a-1), which is the same as option a: za-1.

Explanation
The derivative of ln (cos y) can be found using the chain rule. The derivative of ln u, where u is a function of y, is given by (1/u) * du/dy. In this case, u = cos y. The derivative of cos y with respect to y is -sin y. Therefore, the derivative of ln (cos y) is (-1/cos y) * (-sin y), which simplifies to sec y.

Explanation
The second derivative of y can be found by differentiating the equation twice with respect to x. Using implicit differentiation, we differentiate each term of the equation with respect to x and then solve for y''.

Differentiating the equation once, we get: 8(2y)(dy/dx) + 16y(dy/dx) = 0.
Simplifying, we have: 16y(dy/dx) + 16y(dy/dx) = 0.
Combining like terms, we get: 32y(dy/dx) = 0.

Differentiating again, we get: 32(dy/dx) + 32y(d^2y/dx^2) = 0.
Simplifying, we have: 32y(d^2y/dx^2) = -32(dy/dx).
Dividing both sides by 32y, we get: d^2y/dx^2 = -dy/dx.

Since dy/dx can be written as dy/dx = (dy/dx) / (dx/dx) = (dy/dx) / 1 = dy/dx, the second derivative can be simplified to: d^2y/dx^2 = -dy/dx.

Substituting y' = dy/dx, we have: d^2y/dx^2 = -y'.

Simplifying, we have: d^2y/dx^2 = -y'.

Therefore, the correct answer is (-9/4)y^3.

Explanation
The derivative of 4y is calculated by taking the derivative of y with respect to x. Since there is no x term in the given expression, the derivative of 4y is simply 4. However, the given answer is -4 divided by the square root of (1 - 4x^2). This answer is incorrect as it does not represent the derivative of 4y.