Question

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}}} )

Explanation:

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

The quotient rule is \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\), where \(u = x^{2}\sin(x)\) and \(v=1 + x^{2}\).
First, find \(u^\prime\) using the product rule \((uv)^\prime = u^\prime v+uv^\prime\) (where \(u = x^{2}\), \(v=\sin(x)\)). So \(u^\prime=2x\sin(x)+x^{2}\cos(x)\) and \(v^\prime = 2x\).
Then \(f^\prime(x)=\frac{(2x\sin(x)+x^{2}\cos(x))(1 + x^{2})-x^{2}\sin(x)\cdot2x}{(1 + x^{2})^{2}}\)

$$=\frac{2x\sin(x)+2x^{3}\sin(x)+x^{2}\cos(x)+x^{4}\cos(x)-2x^{3}\sin(x)}{(1 + x^{2})^{2}}=\frac{2x\sin(x)+x^{2}\cos(x)+x^{4}\cos(x)}{(1 + x^{2})^{2}}$$

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

The product rule \((uv)^\prime=u^\prime v + uv^\prime\), where \(u=\sin^{2}(3x)\) and \(v = \sin(4x^{5})\)
For \(u\), using the chain rule: let \(t=\sin(3x)\), \(u = t^{2}\), \(u^\prime=2t\cdot3\cos(3x)=6\sin(3x)\cos(3x)\)
For \(v\), using the chain rule: let \(s = 4x^{5}\), \(v=\sin(s)\), \(v^\prime=20x^{4}\cos(4x^{5})\)
Then \(f^\prime(x)=6\sin(3x)\cos(3x)\sin(4x^{5})+20x^{4}\cos(4x^{5})\sin^{2}(3x)\)

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

Let \(y=\sqrt{u}\), \(u = 1+\sqrt{v}\), \(v=1+\sqrt{w}\), \(w = 1 + x\)
\(y^\prime=\frac{1}{2\sqrt{u}}\cdot u^\prime\), \(u^\prime=\frac{1}{2\sqrt{v}}\cdot v^\prime\), \(v^\prime=\frac{1}{2\sqrt{w}}\cdot w^\prime\)
Since \(w^\prime=1\)
\(v^\prime=\frac{1}{2\sqrt{1 + x}}\), \(u^\prime=\frac{1}{2\sqrt{1+\sqrt{1 + x}}}\cdot\frac{1}{2\sqrt{1 + x}}\)
\(f^\prime(x)=\frac{1}{2\sqrt{1+\sqrt{1+\sqrt{1 + x}}}}\cdot\frac{1}{2\sqrt{1+\sqrt{1 + x}}}\cdot\frac{1}{2\sqrt{1 + x}}=\frac{1}{8\sqrt{1 + x}\sqrt{1+\sqrt{1 + x}}\sqrt{1+\sqrt{1+\sqrt{1 + x}}}}\)

Answer:

(a) \(f^\prime(x)=\frac{2x\sin(x)+x^{2}\cos(x)+x^{4}\cos(x)}{(1 + x^{2})^{2}}\)
(b) \(f^\prime(x)=6\sin(3x)\cos(3x)\sin(4x^{5})+20x^{4}\cos(4x^{5})\sin^{2}(3x)\)
(c) \(f^\prime(x)=\frac{1}{8\sqrt{1 + x}\sqrt{1+\sqrt{1 + x}}\sqrt{1+\sqrt{1+\sqrt{1 + x}}}}\)