Differentiation practice · Mixed difficulty

Option A (-3/5) is correct because after simplifying the inverse cosine function and differentiating, substituting x=-1 yields -3/5.

Option C (0) is correct because the expression simplifies to a constant, and the derivative of a constant is zero.

Option C ((-2,4)) is correct because the given conditions f(1)=3 and f'(1)=0 lead to p=-2 and q=4.

Option B ($4y$) is correct because it is the result of differentiating the given integral equation twice and simplifying.

Option A (6) is correct because after finding the first and second derivatives of y(x) and then adding y and the second derivative, the result is 6.

Option B is correct because it is the result of implicit differentiation of the given equation and solving for dy/dx.

Option A (${\left( {\frac{\pi }{4}} \right)^{1/\sqrt 2 }}.{\left( {\frac{{\sqrt 2 }}{2}\log \frac{4}{\pi } - \frac{{2\sqrt 2 }}{\pi }} \right)}$) is correct because it is the value of the derivative of the function f(x) at x = -π/4, calculated using logarithmic differentiation.

Option C (continuous for all $x$ but not differentiable at $x=0$) is correct because the left-hand limit, right-hand limit, and function value at $x=0$ are all 0, but the left-hand derivative and right-hand derivative are not equal.

Option D ($6 x^2$) is correct because it is the derivative of the determinant function F(x), calculated by differentiating each row separately.

The correct answer is B ($\frac{1}{2}+\frac{\pi}{4}$) because it is the value of the derivative of the given function evaluated at x=1, using the product rule and the derivative of the inverse tangent function.

Option B (1) is correct because when the given expression is simplified using the derivative of the function, all terms cancel out except for a constant which evaluates to 1.

Option C ($f(x) f^{\prime}(n x)$) is correct because, after differentiating the given equation and manipulating the result, we find that $f^{\prime}(x) f(n x) = f(x) f^{\prime}(n x)$.

Option B ($- \frac{x}{y}$) is correct because it is the derivative of y with respect to x, obtained by implicitly differentiating the given equation and using trigonometric identities.

Option D ($\frac{-2}{3}$) is correct because after finding the first and second derivatives and substituting $t = \frac{\pi}{4}$, the expression evaluates to $\frac{-2}{3}$.

Option C (non-existent) is correct because the left-hand derivative and right-hand derivative at x=1 are not equal, so the derivative does not exist.

Option D (All the above) is correct because all three given statements are true based on the given functions and their derivatives.

Option A is correct because it is the result of implicit differentiation of the given equation and solving for dy/dx.

Option A ($f'\left( x \right) = 0$ for all $x \in R$) is correct because the given functional equation implies that $f(x)$ is a constant function, hence its derivative is zero.

Option A ($-4\left\{1+\frac{1}{9}+ \frac{1}{25}+\ldots+\frac{1}{(2 N-1)^2}\right\}$) is correct because it is the result obtained after differentiating the given logarithmic function and summing the series.

The correct answer is C (${\sec ^2}x$) because the expression simplifies to $\tan x$ after applying trigonometric identities, and the derivative of $\tan x$ is ${\sec ^2}x$.

Option A (2) is correct because by differentiating the given equation implicitly and substituting the point (1,0), we can find the value of $\frac{dy}{dx}$.

Option C (-$\frac{2}{3}$) is correct because it is the value of the derivative of the function evaluated at x=0.

Option B (4) is correct because after substituting $x = \frac{1}{2}$, the ratio of the derivatives simplifies to 4.

Option C ($1 + {\left( {g\left( x \right)} \right)^3}$) is correct because it is derived from the inverse function differentiation rule, where $g'(x) = 1/f'(g(x))$.

The correct answer is 1 because after simplifying the given equations and differentiating, ${x^3}y\frac{{dy}}{{dx}}$ equals 1.

Option C ($\frac{1}{{x\left( {2y - 1} \right)}}$) is correct because it is the derivative of the given function after simplifying and differentiating implicitly.

Option A is correct because the derivative is found using the chain rule and trigonometric identities to simplify the expression.

Option C ($x > 0$) is correct because solving the inequality $f'(x) > \phi'(x)$ leads to the condition $x > 0$.

Option B ($- \sqrt {\frac{\mathrm{\pi }}{6}}$) is correct because it is the value of the derivative of the given function evaluated at $x = \frac{{\sqrt \mathrm{\pi } }}{2}$.

Option A (8) is correct because by substituting the functional equation into a simpler form and using the given condition $f(1)=32$, we can find the value of $f(4)$.