Question

Find the first derivative of the following:


`y = x * \sqrt[3]{x}`
then dy/dx=
a)

`\sqrt[3]{x}`
b)

`(4)/(3) \sqrt[3]{x}`
c)

`\frac{4}{ \sqrt[3]{x {}^(2) } }`
d)

`12 \sqrt[3]{x}`

*answer with steps, please​

Answer 1

Consider the given function :

`{:\implies \quad \sf y=x\sqrt[3]{x}}`

Before doing this question let's recall some basic formulae of calculus;

• 
  `{\boxed{\bf{a^(m)\cdot{a^(n)}=a^(m+n)}}}`

• 
  `{\boxed{\bf{(d)/(dx)(x^n)=n{x}^(n-1)}}}`

So, now let's move on to our question ;

`{:\implies \quad \sf y=x\sqrt[3]{x}}`

Can be further written as ;

`{:\implies \quad \sf y=x(x)^(\footnotesize \frac13)}`

Using the 1st identity can be written as ;

`{:\implies \quad \sf y=x^(\footnotesize \frac43)}`

Now, differentiating both sides w.r.t.x ;

`{:\implies \quad \sf (d)/(dx)(y)=(4)/(3)(x)^{\footnotesize (4)/(3)-1}}`

`{:\implies \quad \sf (dy)/(dx)=(4)/(3)(x)^{\footnotesize (4-3)/(3)}}`

`{:\implies \quad \sf (dy)/(dx)=(4)/(3)(x)^{\footnotesize (1)/(3)}}`

`{:\implies \quad \bf \therefore \quad \underline{\underline{(dy)/(dx)=(4)/(3)\sqrt[3]{x}}}}`

Hence, Option B) is correct