#### To determine

**To find:** The composite function in the form f(g(x)) and obtain the derivative of *y*.

#### Answer

The inner function is u=2x3+5 and the outer function is f(u)=u4.

The derivative of *y* is dydx=24x2(2x3+5)3.

#### Explanation

**Given:**

The function is y=(2x3+5)4.

**Formula used:**

**The Chain Rule:**

If *h* is differentiable at *x* and *g* is differentiable at h(x), then the composite function F=g∘h defined by F(x)=g(h(x)) is differentiable at *x* and F′ is given by the product,

F′(x)=g′(h(x))⋅h′(x) (1)

**Derivative Rule:**

(1) Power Rule:ddx(xn)=nxn−1.

(2) Sum Rule: ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)]

**Calculation:**

Let the inner function be u=g(x) and the outer function be y=f(u).

Then, g(x)=2x3+5 and f(u)=u4. That is,

y=(2x3+5)4=f(2x3+5)=f(g(x))

Therefore, y=f(g(x)).

Hence, the inner function is u=2x3+5 and the outer function is f(u)=u4.

Thus, the required form of composite function is f(g(x))=u4.

Obtain the derivative of *y* .

Let h(x)=2x3+5 and g(u)=(u)4 where u=h(x)

Apply the chain rule as shown in equation (1)

y′(x)=g′(h(x))⋅h′(x) (2)

The derivative g′(h(x)) is computed as follows,

g′(h(x))=g′(u)=ddu(g(u))=ddu(u)4

Apply the power rule (2) then substitute u=2x3+5,

g′(h(x))=(4u4−1)=4u3=4(2x3+5)3

The derivative g′(h(x)) is g′(h(x))=4(2x3+5)3.

The derivative of h(x) is computed as follows,

h′(x)=ddx(2x3+5)

Apply the sum rule (2) and the power rule (1),

h′(x)=ddx[2x3]+ddx[5] =2ddx[x3]+[0]=2[3x3−1]+0=6x2

Thus, the derivative of h(x) is h′(x)=6x2.

Substitute 4(2x3+5)3 for g′(h(x)) and 6x2 for h′(x) in equation (2),

g′(h(x))⋅h′(x)=4(2x3+5)3(6x2)=24x2(2x3+5)3

Therefore, The derivative of y=(2x3+5)4 is dydx=24x2(2x3+5)3.