Differentiation practice · Easy

Option A ($\frac{{3 + \sqrt 3 }}{3}$) is correct because it is the only root of the derivative that lies within the specified interval (1, 2).

Option C (0) is correct because after differentiating the given function and substituting x = a, the expression simplifies to zero.

Option C (2) is correct because the function is not differentiable at x = 1 and x = -1 due to the absolute value in the derivative.

Option C (-2) is correct because the derivative of an odd function is an even function, so f'(-3) = f'(3) = -2.

Option C ($6{x^5} + {6^x}\log 6$) is correct because it is the result of applying the power rule and the exponential derivative rule.

The correct answer is None of these because the left-hand derivative and right-hand derivative at x=π/4 are not equal, meaning the function is not differentiable at that point.

Option C is correct because it involves simplifying the expression and then differentiating, evaluating the derivative at the specified point.

Option B ($\frac{{{{\sin }^2}\left( {a + y} \right)}}{{\sin a}}$) is correct because it is the result of differentiating the given equation and solving for $\frac{{dy}}{{dx}}$.

Option A ($\frac{x}{{\sqrt {1 + {x^2}} }}$) is correct because it is the result of simplifying the expression and then differentiating.

Option B (-1) is correct because after differentiating the determinant and substituting x = $\theta$, the expression simplifies to -1.

Option A ($\frac{3}{8}$) is correct because it is the value of the second derivative of y with respect to x at the specified point.

Option A ($\frac{1}{2 \sqrt{3}}$) is correct because it is the value of the derivative of the given function at x = -2, calculated using the chain rule.

Option C (-32) is correct because it is the minimum value of the function g(t) on the interval [-2,1], found by evaluating the function at the endpoints and critical points.

Option C (1) is correct because for $x > 20$, $g(x)$ simplifies to $x - 4$, and its derivative is 1.

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 ($- \frac{1}{4}$) 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 ($y y^{\prime \prime}+\left(y^{\prime}\right)^2+1=0$) 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 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.