problem 1.
differentiate the following functions.
(a) 4pts. ( f(x)=\frac{x^{2} sin (x)}{1+x^{2}} )
(b) 4pts. ( f(x)=sin ^{2}(3 x) sin left(4 x^{5}
ight) )
(c) 4pts. ( f(x)=sqrt{1+sqrt{1+sqrt{1+x}}} )

Question

Accepted Answer

# Explanation:
## Step1: Differentiate $f(x)=\frac{x^{2}\sin(x)}{1 + x^{2}}$ using the quotient rule
The quotient rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = x^{2}\sin(x)$, $u^\prime=2x\sin(x)+x^{2}\cos(x)$ (using the product rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = x^{2}$, $u^\prime=2x$ and $v=\sin(x)$, $v^\prime=\cos(x)$), and $v = 1 + x^{2}$, $v^\prime=2x$.

$$
\begin{align*}
f^\prime(x)&=\frac{(2x\sin(x)+x^{2}\cos(x))(1 + x^{2})-x^{2}\sin(x)\cdot(2x)}{(1 + x^{2})^{2}}\
&=\frac{2x\sin(x)+2x^{3}\sin(x)+x^{2}\cos(x)+x^{4}\cos(x)-2x^{3}\sin(x)}{(1 + x^{2})^{2}}\
&=\frac{2x\sin(x)+x^{2}\cos(x)+x^{4}\cos(x)}{(1 + x^{2})^{2}}
\end{align*}
$$

## Step2: Differentiate $f(x)=\sin^{2}(3x)\sin(4x^{5})$ using the product rule and chain rule
The product rule: if $y = u\cdot v$, then $y^\prime=u^\prime v+uv^\prime$. For $u=\sin^{2}(3x)$, using the chain rule $(u = g(h(x)))$, $u^\prime = 2\sin(3x)\cos(3x)\cdot3=3\sin(6x)$ (since $\sin(2\alpha)=2\sin\alpha\cos\alpha$). For $v=\sin(4x^{5})$, $v^\prime=\cos(4x^{5})\cdot20x^{4}$

$$
\begin{align*}
f^\prime(x)&=3\sin(6x)\sin(4x^{5})+\sin^{2}(3x)\cdot20x^{4}\cos(4x^{5})
\end{align*}
$$

## Step3: Differentiate $f(x)=\sqrt{1+\sqrt{1+\sqrt{1 + x}}}$ using the chain rule
Let $y=\sqrt{u}$, $u = 1+\sqrt{v}$, $v=1+\sqrt{w}$, $w = 1 + x$.

$y^\prime=\frac{1}{2\sqrt{u}}\cdot u^\prime$, $u^\prime=\frac{1}{2\sqrt{v}}\cdot v^\prime$, $v^\prime=\frac{1}{2\sqrt{w}}\cdot w^\prime$, and $w^\prime = 1$

$$
\begin{align*}
f^\prime(x)&=\frac{1}{2\sqrt{1+\sqrt{1+\sqrt{1 + x}}}}\cdot\frac{1}{2\sqrt{1+\sqrt{1 + x}}}\cdot\frac{1}{2\sqrt{1 + x}}\
&=\frac{1}{8\sqrt{1 + x}\sqrt{1+\sqrt{1 + x}}\sqrt{1+\sqrt{1+\sqrt{1 + x}}}}
\end{align*}
$$

# Answer:
(a) $f^\prime(x)=\frac{2x\sin(x)+x^{2}\cos(x)+x^{4}\cos(x)}{(1 + x^{2})^{2}}$

(b) $f^\prime(x)=3\sin(6x)\sin(4x^{5})+20x^{4}\sin^{2}(3x)\cos(4x^{5})$

(c) $f^\prime(x)=\frac{1}{8\sqrt{1 + x}\sqrt{1+\sqrt{1 + x}}\sqrt{1+\sqrt{1+\sqrt{1 + x}}}}$

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QUESTION IMAGE

Question

Explanation:

Step1: Differentiate \(f(x)=\frac{x^{2}\sin(x)}{1 + x^{2}}\) using the quotient rule

Step2: Differentiate \(f(x)=\sin^{2}(3x)\sin(4x^{5})\) using the product rule and chain rule

Step3: Differentiate \(f(x)=\sqrt{1+\sqrt{1+\sqrt{1 + x}}}\) using the chain rule

Answer: