How do you differentiate $y = e^{2 x^{3}}$ $y=e^(2x^3)$ ?

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How do you differentiate $y = e^{2 x^{3}}$ $y=e^(2x^3)$ ?

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Answer 1 · 2016-10-05T09:50:22

$\frac{d y}{d x} = 6 x^{2} e^{2 x^{3}}$ $(dy)/(dx)=6x^2e^(2x^3)$

Explanation:

We use **Chain Rule ** here.

In order to differentiate a function of a function, say $y, = f (g (x))$ $y, =f(g(x))$ , where we have to find $\frac{d y}{d x}$ $(dy)/(dx)$ , we need to do substitute $u = g (x)$ $u=g(x)$ , which gives us $y = f (u)$ $y=f(u)$ .

The Chain Rule states that $\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$ $(dy)/(dx)=(dy)/(du)xx(du)/(dx)$ .

In fact if we have something like $y = f (g (h (x)))$ $y=f(g(h(x)))$ , we can have $\frac{d y}{d x} = \frac{d y}{d f} \times \frac{d f}{d g} \times \frac{d g}{d h}$ $(dy)/(dx)=(dy)/(df)xx(df)/(dg)xx(dg)/(dh)$

Here we have $y = e^{2 x^{3}}$ $y=e^(2x^3)$

Hence $\frac{d y}{d x} = \frac{d (e^{2 x^{3}})}{d (2 x^{3})} \times \frac{d (2 x^{3})}{d x}$ $(dy)/(dx)=(d(e^(2x^3)))/(d(2x^3))xx(d(2x^3))/(dx)$

= $e^{2 x^{3}} \times 2 \times 3 x^{2}$ $e^(2x^3)xx2xx3x^2$

= $6 x^{2} e^{2 x^{3}}$ $6x^2e^(2x^3)$

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How do you differentiate $y = e^{2 x^{3}}$ $y=e^(2x^3)$ ?

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How do you differentiate $y = e^{2 x^{3}}$ $y=e^(2x^3)$ ?