Differentiation practice · Mixed difficulty

Option B ($x=-4.9 t^2+19.6(0 \leq t \leq 2)$) is correct because it accurately describes the ball's position as a function of time under constant gravitational acceleration, with the correct initial height and time interval.

Option D is correct. By differentiating f(x) twice and substituting into the given equation, we can solve for g(x) and find that it equals $\cos ^{2} x \sin (\sin x)$.

Option D ($k f(x)$) is correct because it is the result of applying Leibniz's rule for differentiating under the integral sign and simplifying the expression.

Option A (512) is correct because it is the value obtained after calculating the derivative of the determinant and substituting x = π/4.

Option D ($- \frac{1}{4}$) is correct because after simplifying the expression using inverse trigonometric properties and then differentiating, y'(1) equals $-\frac{1}{4}$.

Option A ($\frac{1}{2 \sqrt{1+\sqrt{2}}}(\frac{3+4 \sqrt{2}}{4 \sqrt{2}})$) is correct because it follows from applying the chain rule to find the derivative and then substituting x=1.

Option C (2) is correct because the minimum value of the function occurs at x=$\pi$/4, where the function value is 2.

Option B is correct because the expression is simplified using trigonometric identities and then differentiated, resulting in cosec x.

Option B (Constant =1) is correct because after simplification, the derivative of y with respect to x is equal to 1.

Option D ($\sqrt{\pi}$) is correct because it is the value obtained after differentiating the given function and substituting the given value of x.

Option B ($y y^{\prime \prime}+\left(y^{\prime}\right)^2+1=0$) is correct because it is the result of differentiating the given equation $x^2 + y^2 = 1$ twice with respect to x.

Option B (0) is correct because after differentiating the given equation with respect to x, it simplifies to 1 + dy/dx = 0, and the second derivative is therefore 0.

Option A ($\frac{1}{\sqrt{2}}$) is correct because it is the value of the derivative of the given function evaluated at x=1.

Option A is correct because the second derivative is found by implicit differentiation of the given equation, resulting in a constant value.

Option D (-25y) is correct because after differentiating the given equations and substituting, the expression simplifies to -25y.

Option B ($\frac{-1}{(1+x)^2}$) is correct because simplifying the given equation and then differentiating yields dy/dx = $\frac{-1}{(1+x)^2}$.

Option A is correct because it is the result of logarithmic differentiation and simplification.

Option C ($- {e^{y - x}}$) is correct because it is the derivative of y with respect to x after implicit differentiation and simplification.

Option B ($\frac{{ - 1}}{{{{(1 + x)}^2}}}$) is correct because it is the result of differentiating the simplified equation after eliminating y.

Option D ($F'\left[ {f\left\{ {\phi \left( x \right)} \right\}} \right]f'\left\{ {\phi \left( x \right)} \right\}\phi '\left( x \right)$) is correct because it correctly applies the chain rule multiple times.

Option D ($\frac{{(-d^{2} y / d x^{2})}}{{(d y / d x)^{2}}}$) is correct because it represents the second derivative of x with respect to y, which is the negative of the second derivative of y with respect to x divided by the square of the first derivative of y with respect to x.

Option D ($225 y$) is correct. By differentiating the given equation twice and substituting, we find that $(x^{2}-1) \frac{d^{2} y}{dx^{2}}+x \frac{dy}{dx} = 225y$.

Option D, a( \fracb^r - 1b - 1 ) + b^rx$a\left( {\frac{{{{b^r} - 1}}{{{{b - 1}}}}} \right) + {{b^r}x}$, is correct because it represents the general form of the function f(x) repeated r times, derived by iteratively applying the function to itself.

Option A (4) is correct. By differentiating the given function twice and substituting into the given differential equation, the value of \$\lambda$\$ is found to be 4.

Option C ($\frac{27}{4} c^3$) is correct because it is the value of $a^2b$ calculated from the given curve equation and tangent properties.

Option C (1) is correct because after simplifying the function and finding its derivative, the value of $10f'(102^+)$ is 1.

Option B ($2a-4$) is correct because it is the result of applying L'Hopital's rule or algebraic manipulation to evaluate the limit using the given derivative and function value.

Option B is correct. gof is differentiable at x=0 and its derivative is continuous at that point, but gof is not twice differentiable at x=0.

Option C ($\cos x$) is correct because it is the result of differentiating the given equation and solving for (2y-1)dy/dx.

Option A is correct because $(2+a)^2 f^{\prime \prime}(1)+(2-a)^2 f^{\prime \prime}(-1)=0$ is derived from calculating the first and second derivatives of the given function and substituting x=1 and x=-1.