Question

find derivative of:
$\frac{(x {}^{2} - 1)}{x}$
using quotient rule..​

Answer 1

Step-by-step explanation:

Given that,

y=\dfrac{x^5-\cos x}{\sin x}y=

sinx

x

5

−cosx

From this, we get,

\begin{gathered}\rightarrow\ f(x)=x^5-\cos x\\ \\ \rightarrow\ g(x)=\sin x\end{gathered}

→ f(x)=x

5

−cosx

→ g(x)=sinx

Now we find f'(x) and g'(x).

\begin{gathered}\rightarrow\ f'(x)=5x^4+\sin x\\ \\ \rightarrow\ g'(x)=\cos x\end{gathered}

→ f

′

(x)=5x

4

+sinx

→ g

′

(x)=cosx

So,

\begin{gathered}\begin{aligned}y'&=\ \ \frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}\\ \\ y'&=\ \ \frac{\sin x(5x^4+\sin x)-(x^5-\cos x)\cos x}{\sin^2x}\\ \\ y'&=\ \ \frac{5x^4\sin x+\sin^2x-x^5\cos x+\cos^2x}{\sin^2x}\\ \\ \\ \Large \text{$y'$}&=\ \ \Large \text{$\frac{-x^5\cos x+5x^4\sin x+1}{\sin^2x}$}\end{aligned}\end{gathered}

y

′

y

′

y

′

y

′

=

(g(x))

2

g(x)f

′

(x)−f(x)g

′

(x)

=

sin

2

x

sinx(5x

4

+sinx)−(x

5

−cosx)cosx

=

sin

2

x

5x

4

sinx+sin

2

x−x

5

cosx+cos

2

x

=

sin

2

x

−x

5

cosx+5x

4

sinx+1

Hence, Derived

Answer 2

answer is 6 / 2 is correct answer