Question

1. calcule a 1ª derivada das seguintes funções

a. $f(x)=sin3x$
b. $h(x)=cos3x^{2}$
c. $g(x)=sin^{2}x$
d. $m(x)=x^{x^{2}+8x + 7}$
e. $a(x)=ln5x^{2}+3$

Explanation:

Step1: For $f(x)=\sin3x$, use chain - rule

Let $u = 3x$, then $y=\sin u$. $\frac{du}{dx}=3$, $\frac{dy}{du}=\cos u$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$, so $f^\prime(x)=\cos(3x)\cdot3 = 3\cos3x$.

Step2: For $h(x)=\cos(3x^{2})$, use chain - rule

Let $u = 3x^{2}$, then $y=\cos u$. $\frac{du}{dx}=6x$, $\frac{dy}{du}=-\sin u$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$, so $h^\prime(x)=-\sin(3x^{2})\cdot6x=-6x\sin(3x^{2})$.

Step3: For $g(x)=\sin^{2}x$, use chain - rule

Let $u=\sin x$, then $y = u^{2}$. $\frac{du}{dx}=\cos x$, $\frac{dy}{du}=2u$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$, so $g^\prime(x)=2\sin x\cos x=\sin2x$.

Step4: For $m(x)=x^{2}+8x + 7$, use power - rule

$m^\prime(x)=(x^{2})^\prime+(8x)^\prime+(7)^\prime$. According to the power - rule $(x^{n})^\prime=nx^{n - 1}$, so $m^\prime(x)=2x + 8$.

Step5: For $a(x)=\ln(5x^{2}+3)$, use chain - rule

Let $u = 5x^{2}+3$, then $y=\ln u$. $\frac{du}{dx}=10x$, $\frac{dy}{du}=\frac{1}{u}$. By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$, so $a^\prime(x)=\frac{10x}{5x^{2}+3}$.

Answer:

A. $f^\prime(x)=3\cos3x$
B. $h^\prime(x)=-6x\sin(3x^{2})$
C. $g^\prime(x)=\sin2x$
D. $m^\prime(x)=2x + 8$
E. $a^\prime(x)=\frac{10x}{5x^{2}+3}$