12. qual é a primeira derivada da função $f(x)=ln x$\na. $-\frac{1}{x}$ b. $-\frac{1}{x^{2}}$ c…

12. qual é a primeira derivada da função $f(x)=ln x$\na. $-\frac{1}{x}$ b. $-\frac{1}{x^{2}}$ c. $\frac{1}{x^{2}}$ d. $\frac{1}{x}$\n13. considere a função $f(x)$ representada pelo gráfico. em que intervalo da função $f(x)$, a primeira derivada é negativa?\na $-infty;-2$\nb $-2;2$\nc $-1;1$\nd $0;2$\n14. qual é a primeira derivada da função $f(x)=2ln x$\na $-\frac{1}{x}$ b $2x^{-1}$ c. $\frac{1}{x^{2}}$ d. $\frac{1}{x}$\n15. qual é a 1ª derivada da função $f(x)=cos(x^{2}+1)$\na. $-sin(x^{2}+1)$ b. $-2xsin(x^{2}+1)$ c. $2xsin(x^{2}-1)$ d. $sin(x^{2}+1)$\n16. qual é a primeira derivada de $f(x)=x^{3}.sin x?$\na $3x^{2}.cos x$ b $6xsin x$ c $x^{2}.(sin x + xcos x)$ d $x^{2}.(3sin x+xcos x)$\n17. em que intervalo a função $f(x)=x^{3}-3x$ é decrescente?\na $-1;1$ b $-infty;-1$ b $-1;2$ d $1;+infty$\n18. qual é a primeira derivada da função $f(x)=\frac{7 - x}{(x - 1)^{2}}$ não - é - derivável\na $f(x)=-\frac{7}{(x - 1)^{2}}$ b $f(x)=\frac{7}{(x - 1)^{2}}$ c $f(x)=\frac{5}{(x - 1)^{2}}$ d $f(x)=\frac{8}{(x - 1)^{2}}$\n19. qual é a segunda derivada da função $f(x)=e^{x}(x + 1)$?\na $xe^{x}(x + 2)$ b. $ae^{x}(x + 2)$ c $xe^{x}(x - 2)$ d $e^{x}(x - 2)$\n20 a função inversa da derivada de $f(x)=\frac{x}{3}$ é:\na. $\frac{1}{3}$ b. $\frac{x}{3}$ c. $3x$ d. $e^{x}$

12. qual é a primeira derivada da função $f(x)=ln x$\na. $-\frac{1}{x}$ b. $-\frac{1}{x^{2}}$ c. $\frac{1}{x^{2}}$ d. $\frac{1}{x}$\n13. considere a função $f(x)$ representada pelo gráfico. em que intervalo da função $f(x)$, a primeira derivada é negativa?\na $-infty;-2$\nb $-2;2$\nc $-1;1$\nd $0;2$\n14. qual é a primeira derivada da função $f(x)=2ln x$\na $-\frac{1}{x}$ b $2x^{-1}$ c. $\frac{1}{x^{2}}$ d. $\frac{1}{x}$\n15. qual é a 1ª derivada da função $f(x)=cos(x^{2}+1)$\na. $-sin(x^{2}+1)$ b. $-2xsin(x^{2}+1)$ c. $2xsin(x^{2}-1)$ d. $sin(x^{2}+1)$\n16. qual é a primeira derivada de $f(x)=x^{3}.sin x?$\na $3x^{2}.cos x$ b $6xsin x$ c $x^{2}.(sin x + xcos x)$ d $x^{2}.(3sin x+xcos x)$\n17. em que intervalo a função $f(x)=x^{3}-3x$ é decrescente?\na $-1;1$ b $-infty;-1$ b $-1;2$ d $1;+infty$\n18. qual é a primeira derivada da função $f(x)=\frac{7 - x}{(x - 1)^{2}}$ não - é - derivável\na $f(x)=-\frac{7}{(x - 1)^{2}}$ b $f(x)=\frac{7}{(x - 1)^{2}}$ c $f(x)=\frac{5}{(x - 1)^{2}}$ d $f(x)=\frac{8}{(x - 1)^{2}}$\n19. qual é a segunda derivada da função $f(x)=e^{x}(x + 1)$?\na $xe^{x}(x + 2)$ b. $ae^{x}(x + 2)$ c $xe^{x}(x - 2)$ d $e^{x}(x - 2)$\n20 a função inversa da derivada de $f(x)=\frac{x}{3}$ é:\na. $\frac{1}{3}$ b. $\frac{x}{3}$ c. $3x$ d. $e^{x}$

Answer

12.

Explanation:

Step1: Recall derivative formula

The derivative of $y = \ln x$ is $y^\prime=\frac{1}{x}$ according to the derivative formula of logarithmic - function $(\ln x)^\prime=\frac{1}{x}$.

Answer:

D. $\frac{1}{x}$

13.

Explanation:

Step1: Analyze the relationship between derivative and monotonicity

The first - derivative of a function is negative when the function is decreasing. By observing the graph, we can see that the function is decreasing in the interval $(-\infty,-2)$.

Answer:

A. $]-\infty,-2[$

14.

Explanation:

Step1: Use the constant - multiple rule of derivatives

Since $f(x)=2\ln x$, according to the constant - multiple rule of derivatives $(cf(x))^\prime = cf^\prime(x)$ and $(\ln x)^\prime=\frac{1}{x}$, then $f^\prime(x)=2\times\frac{1}{x}=\frac{2}{x}=2x^{-1}$.

Answer:

B. $2x^{-1}$

15.

Explanation:

Step1: Use the chain - rule

Let $u = x^{2}+1$, then $y=\cos u$. According to the chain - rule $y^\prime_x=y^\prime_u\cdot u^\prime_x$. We know that $y^\prime_u=-\sin u$ and $u^\prime_x = 2x$. Substituting $u = x^{2}+1$ back, we get $y^\prime=-2x\sin(x^{2}+1)$.

Answer:

B. $-2x\sin(x^{2}+1)$

16.

Explanation:

Step1: Use the product - rule

The product - rule is $(uv)^\prime = u^\prime v+uv^\prime$. Let $u = x^{3}$ and $v=\sin x$. Then $u^\prime = 3x^{2}$ and $v^\prime=\cos x$. So $f^\prime(x)=3x^{2}\sin x+x^{3}\cos x=x^{2}(3\sin x + x\cos x)$.

Answer:

D. $x^{2}(3\sin x + x\cos x)$

17.

Explanation:

Step1: Find the derivative of the function

First, find the derivative of $f(x)=x^{3}-3x$, $f^\prime(x)=3x^{2}-3 = 3(x^{2}-1)=3(x + 1)(x - 1)$. Then find where $f^\prime(x)<0$. Solving the inequality $3(x + 1)(x - 1)<0$, we get $-1<x<1$.

Answer:

A. $]-1;1[$

18.

Explanation:

Step1: Use the quotient - rule

The quotient - rule is $(\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Let $u = 7 - x$ and $v=x - 1$, then $u^\prime=-1$ and $v^\prime = 1$. So $f^\prime(x)=\frac{-1\times(x - 1)-(7 - x)\times1}{(x - 1)^{2}}=\frac{-x + 1-7 + x}{(x - 1)^{2}}=\frac{-6}{(x - 1)^{2}}$.

Answer:

There is no correct option in the given choices.

19.

Explanation:

Step1: Use the product - rule to find the first - derivative

Let $u = e^{x}$ and $v=x + 1$, then $f(x)=uv$. By the product - rule $f^\prime(x)=u^\prime v+uv^\prime=e^{x}(x + 1)+e^{x}=e^{x}(x + 2)$.

Step2: Use the product - rule to find the second - derivative

Let $u = e^{x}$ and $v=x + 2$, then $f^{\prime\prime}(x)=u^\prime v+uv^\prime=e^{x}(x + 2)+e^{x}=e^{x}(x+3)$. There is no correct option in the given choices.

20.

Explanation:

Step1: Find the derivative of $f(x)=\frac{x^{3}}{3}$

$f^\prime(x)=x^{2}$. The inverse function of $y = x^{2}(x\geq0)$ (assuming we consider the principal - value range) has an inverse relationship. The antiderivative of $x^{2}$ is $\frac{1}{3}x^{3}+C$. If we consider the inverse of the derivative operation, the function whose derivative is $x^{2}$ is $\frac{1}{3}x^{3}$.

Answer:

A. $\frac{1}{3}x^{3}$ (It should be noted that the original option A is written as $\frac{1}{3}$, which may be a mis - writing. It should be $\frac{1}{3}x^{3}$)