Differentiation practice · Medium

The correct answer is D ($\cos ^{2} x \sin (\sin x)$) because it is the function f(x) that satisfies the given differential equation after finding the first and second derivatives of y.

Option A (-1) is correct because after finding dy/dx in terms of theta, substituting theta = 3pi/4 yields -1.

Option B (-n⁴ f(x)) is correct because it is the second derivative of the given function, calculated using Euler's formula and differentiation.

Option D (1) is correct because the given condition implies that the derivative of $f(x)$ is zero everywhere, making $f(x)$ a constant function. Since $f(0)=1$, $f(x)=1$ for all $x$, and the integral of $f^2(x)$ from 0 to 1 is 1.

Option C ($2 a f(a)-a^2 f^{\prime}(a)$) is correct because it is the result of applying the definition of the derivative and algebraic manipulation to evaluate the limit.

Option B ($\frac{y}{x}$) is correct because implicit differentiation and simplification lead to this result.

Option A is correct because the points where g(x) is non-differentiable are the points where the absolute value functions inside g(x) change sign, which are 5, 10, and 15.

Option C ($- \frac{{12}}{5}$) is correct because the rate of change is calculated by finding the derivatives of both functions and then dividing them, evaluated at x=3.

Option C ($\operatorname{cosec} x$) is correct because by applying the chain rule and simplifying the derivative of $\log |\tan \frac{x}{2}|$, we obtain $\operatorname{cosec} x$.

Option C ($\frac{\sqrt{3}-1}{2}$) is correct because it is the value of the derivative of the function at the given point after considering the absolute values.

Option D ($\frac{2^{\log x} \cdot \log 2}{x}$) is correct because it is the result of applying the chain rule and derivative of exponential functions.

Option A (-1) is correct because it is the value of y'(1) obtained by implicit differentiation and substituting x=1 and y=pi/2.

Option C ($f(x)$ is invertible) is correct because the derivative is always positive, indicating that the function is strictly increasing and therefore invertible.

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 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 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 D is correct because it is the result of applying the chain rule and evaluating the derivative at x=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 $\frac{1}{e}$.

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.