Differentiation practice · Mixed difficulty

Option A is correct because it is the result of applying the quotient rule and chain rule to differentiate the given logarithmic function.

Option B is correct because both the assertion and the reason are correct statements, but the reason does not provide a direct explanation for the assertion.

Option B ($\frac{1}{x}$ for $\left| x \right| > 1$ and $\frac{{ - 1}}{x}$ for $\left| x \right| < 1$) is correct because it correctly describes the derivative of the given function for different ranges of x.

Option D ($1+\{g(x)\}^5$) is correct because it is derived from differentiating the inverse function relationship and substituting the given derivative of f(x).

Option D ($\frac{1}{2 e}$) is correct because after simplifying the function and finding its derivative, the value of $f'(e)$ is $\frac{1}{2e}$.

Option B ($2^n$) is correct because the given expression is the binomial expansion of $(1+1)^n$ evaluated at $x=0$.

Option C ($n(n - 1)2^{n - 2}$) is correct because the given series is the binomial expansion of $(1 + x)^n$, and its second derivative evaluated at x=1 is $n(n - 1)2^{n - 2}$.

The correct answer is B ($\frac{{\left| {y''} \right|}}{{\sqrt {{{\left( {1 + {{y'}^2}} \right)}^3}} }}$) because it represents the radius of curvature $k = \frac{1}{a}$ derived from the given equation and its derivatives.

Option D (All of the above) is correct because all the given relationships between the derivatives of f(x) and g(x) are valid based on the given equations.

The correct answer is A (-1) because it is the value of the derivative y'(1) obtained after implicit differentiation and substitution.

Option A is correct because the derivative is found by logarithmic differentiation, simplifying the expression before differentiating.

Option B (${y^2}$) is correct because after taking logarithms and differentiating, the expression simplifies to $y^2$.

Option D (if $f$ is continuously differentiable) is correct because the Intermediate Value Theorem can be applied to the derivative of a continuously differentiable function.

Option B ($\frac{{1}}{{2+[g(x)-x]^2}}$) is correct because by differentiating $f(g(x)) = x$ and using trigonometric identities, we arrive at the given expression.

Option D $\frac{{5}}{{4}}$ is correct because it is the x-coordinate where the second derivative of the function is zero and the third derivative is non-zero.

Option C (-6) is correct because it is the value of $f'(5)$ found using the given functional equation and derivative at 0.

Option A (3) is correct because the equation can be rewritten as a trigonometric function with a range that allows for only three integer values of k.

Option C (1) is correct because the second derivative of an even function is also even, hence $f''(-x) = f''(x)$.

Option C is correct because the function is continuous for all x but not differentiable at x=0 due to unequal left-hand and right-hand derivatives.

Option B ($\frac{3}{(1-x)^2}$) is correct because it is the derivative of the inverse function after simplification.

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

Option D is correct because it is the result of applying the chain rule and evaluating the derivative at x=1.

Option D, $f(1)+f(3)=f(0)$, is false because the correct relationship between the function values is $f(0)+f(2)=f(1)$.

The correct answer is A ($\frac{{x^2 y^2+1}}{{2 x^2}}$) because after differentiating the given equation implicitly and substituting the value of $\log x$ from the original equation, we arrive at this expression.

Option A (${e^{{e^x}}}$) is incorrect. Option B (${e^x}$) is incorrect. Option C (${e^{{e^x} + x}}$) is correct because it is the derivative of the inverse function g(x) found by using the relationship between f(x) and g(x). Option D (None of these) is incorrect.

Option A (- 1) is correct because after taking the logarithm and differentiating implicitly, the derivative at the point (2, 2) is -1.

Option C (Is equal to $\frac{1}{e}$) is correct because after simplification and differentiation, substituting x = e yields the value $\frac{1}{e}$.

Option A (3) is correct because by differentiating F(x) and substituting the given conditions, the value of k/8 is found to be 3.

Option B (1) is correct because the function simplifies to y = x + constant, making the derivative equal to 1.

The correct answer is C ($1-e^{-1}$) because it is the largest possible value of f(1) derived from the given differential inequality and initial condition.