Differentiation practice · Easy

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

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

Option A (($\pi$ + 1)/4) is correct because, after simplifying f'(x) and integrating, using the initial condition f(0) = 0, we find f(1) = ($\pi$ + 1)/4.

Option C ($\frac{{1-x}}{{x}}$) is correct because the derivative $\frac{{dy}}{{dx}}$ is found by implicit differentiation after taking the logarithm of both sides of the given equation.

Option A ($\frac{1}{2}+\frac{\pi}{4}$) is correct because it is the value of the derivative of $f(x)=x \tan ^{-1} x$ evaluated at $x=1$.

Option D (${\left( { - 1} \right)^n}\left( {n - 1} \right)!$) is correct because it is obtained by differentiating the given product and substituting $x = -1$.

Option C ($\ - \frac{1}{{14}}$) is correct because after solving for $f(x)$ and substituting it into $y = x^2f(x)$, the derivative at $x = -1$ is $-\frac{1}{14}$.

Option B (1) is correct because the derivative of f(x) = x+2 is always 1, regardless of the input.

Option A ($\frac{{1}}{{2}}+\frac{{\pi}}{{4}}$) is correct because it is the value of the derivative of $f(x)=x \tan^{-1}x$ at $x=1$.

Option B is correct because the limit $\lim_{h \rightarrow 0} \frac{f(h)-f(0)}{\sqrt{|h|}}$ exists and is finite for $f(x)=x^{2 / 3}$.

Option D ($|\sec \theta|$) is correct because differentiating and simplifying the expression leads to the absolute value of secant.

Option C ($\frac{1}{{\sqrt {1 - {x^2}} }} - \frac{1}{{2\sqrt {x\left( {1 - x} \right)} }}$) is correct because it represents the derivative of the given expression after applying trigonometric substitution and simplification.

The correct answer is B ($\left( {0,\\,\frac{{2\pi }}{3}} \right)$) because the normal to the curve at $x = \frac{\pi }{6}$ passes through this point.

Option D ($f\left( 6 \right) \ge 8$) is correct because it is derived from the Mean Value Theorem, using the given condition on the derivative and the value of the function at x=1.

Option A (0) is correct because after simplification, the function f(x) becomes a constant, and the derivative of a constant is zero.

Option B ($\frac{{\left( {2e - 1} \right)}}{{2\sqrt {4 + {e^2}} }}$) is correct because it is the value of the derivative at x=e, obtained by implicit differentiation and substitution.

Option C (12) is correct because, by the chain rule, $h'(x) = f'(g(x)) * g'(x)$, so $h'(3) = f'(g(3)) * g'(3) = f'(7) * 6 = 2 * 6 = 12$.

Option B ($\frac{2 f(x) g(x)}{1+(g(x))^2}$) is correct because differentiating $\ln \left(1+(g(x))^2\right)$ gives the expression.

Option A ($x^{\log x}(\frac{{2 \log x}}{{x}})$) is correct because it's the result of applying logarithmic differentiation to the given function.