Differentiation practice · Medium

Option A is correct because it is the result of implicit differentiation and substitution to find y'(0).

Option D (None of these) is correct because a function can have a minimum or maximum even if its derivative is not zero at a point (e.g., at a cusp or endpoint).

Option B is correct because substituting x=0 into the differentiated equation yields dy/dx = 1.

Option C is correct because at x=pi/4, the derivative simplifies to 2 after substituting the value.

Option A ($\frac{1}{{\sqrt {1 - {x^2}} }}$) is correct because after simplification using trigonometric identities, the derivative can be easily found.

Option B ($- \frac{y}{x}$) is correct because after using the identity $\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}$, the derivative simplifies to $- \frac{y}{x}$.

Option D (All of these) is correct because when we solve for the coefficients of the cubic polynomial, all the given relations hold true.

Option C ($-\frac{{4}}{{3}}$) is correct because it is the value of $\frac{{dy}}{{dx}}$ calculated using parametric differentiation at $t=2$.

Option D ($- \frac{1}{2}$) is correct because after simplifying the expression inside the inverse tangent and differentiating, the result is $- \frac{1}{2}$.

Option B ($- 3$) is correct because the derivative of $f(x) = 3|2 + x|$ at $x_0 = -3$ is -3, as the function is defined as $f(x) = -3(2+x)$ for $x < -2$.

Option D ($6 f(x) g(x)$) is correct because it is the result of differentiating $T(x)$ using the product rule and substituting the given derivatives of $f(x)$ and $g(x)$.

Option C, $\log x.{\left( {1 + \log x} \right)^{ - 2}}$, is correct because it represents the derivative dy/dx obtained by implicit differentiation and simplification of the given equation ${x^y} = {e^{x - y}}$.

Option A is correct because it accurately applies the chain rule and logarithmic differentiation to find the derivative of the given function.

Option B ($\frac{{{y^2}}}{{x(1 - y\log x)}}$) is correct because it results from applying logarithmic differentiation to the given function and solving for $\frac{{dy}}{{dx}}$.

Option B ($n - 1$) is correct because the derivative of a polynomial of degree $n$ is a polynomial of degree $n-1$.

Option D (None of these) is correct because the nth derivative of the product of cosine functions involves a sum of cosine functions with arguments adjusted by nπ/2, and the given options do not match the correct form.

Option A ($a=-3 / 4, b=-1 / 8$) is correct because these values satisfy the conditions for the function to have extrema at x=1 and x=3.

Option C ($1 + {\{\phi (x)\}^5}$) is correct because it is derived from the inverse function differentiation rule.

Option B ($\frac{1}{1+x^2}$) is correct because after substituting $x = \tan \theta$ and simplifying the expression, the derivative is found to be $\frac{1}{1+x^2}$.

Option B ($\frac{1}{2}$) is correct because it is the value of the derivative $f'(2)$ after solving for $f(x)$ and differentiating.

Option B ($2^n$) is correct because the given expression is the binomial expansion of $(1+1)^n$, which equals $2^n$.

Option A ($\frac{2}{3}$) is correct because by substituting $x = \sin \theta$, the expression simplifies, and the derivative becomes a simple ratio.

Option C (8) is correct because it follows from the chain rule applied to composite functions.

Option A ($2 x^{\left(\log _a x-1\right)} \cdot \log _e x$) is correct because by applying logarithmic differentiation, we can find the derivative of the given function.

Option C ($\frac{y}{x}$) is correct because it is the derivative dy/dx obtained after simplifying the given equation using trigonometric identities and differentiating implicitly.

Option D (Does not exist) is correct because the left-hand derivative and right-hand derivative at $x = 0$ are not equal.

Option A ($\frac{\log _2 \mathrm{e}}{\mathrm{x} \ln \mathrm{x}}$) is correct because it is the result of applying the chain rule and change of base formula to differentiate the given logarithmic function.

Option C ($1 + {\left[ {\phi \left( x \right)} \right]^5}$) is correct because it is the derivative of the inverse function, calculated using the given derivative of the original function.

Option C (100) is correct because f'(1) = 100 and f'(0) = 1, so f'(1) = 100f'(0).

Option D (8) is correct because by using the given functional equation and derivative information, we can find f'(4).