Function of a function can be solved using the chain rule.

Here's a step-by-step guide on how to solve differentiation of a function of a function using the chain rule:

**1. Identify the inner and outer functions:**

Let y = f(g(x)), where f and g are functions. Identify g(x) as the inner function and f(u) as the outer function.

**2. Differentiate the outer function with respect to the inner function:**

Find the derivative of f(u) with respect to u, denoted as f'(u).

**3. Differentiate the inner function with respect to x:**

Find the derivative of g(x) with respect to x, denoted as g'(x).

**4. Multiply the two derivatives together:**

Multiply f'(u) and g'(x) together to get the derivative of the composite functions.

**5. Replace u with g(x) in the final expression:**

Substitute g(x) for u in the resulting expression to get the final derivative of y with respect to x.

**1. Differentiate the function f(x) = (3x^2 + 1)^3.**

**Solution**:

To differentiate a function of a function, we can use the chain rule. Let u = 3x^2 + 1. Then, f(x) = u^3.

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 6x.

Now, applying the chain rule, we have:

\frac{df}{dx} = \frac{d(u^3)}{du} \cdot \frac{du}{dx}

\frac{df}{dx} = 3u^2 \cdot 6x

\frac{df}{dx} = 18x(3x^2 + 1)^2.

**2. Find the derivative of g(x) = e^{4x^2}.**

**Solution: **

To differentiate a function of a function, we can use the chain rule. Let u = 4x^2, then g(x) = e^u.

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 8x.

Then, applying the chain rule, we have:

\frac{dg}{dx} = e^u \cdot 8x

\frac{dg}{dx} = 8x e^{4x^2}.

**3. Differentiate the function h(x) = \sqrt{5x^3 + 2}.**

**Solution**:

To differentiate a function of a function, we can use the chain rule. Let u = 5x^3 + 2, then h(x) = \sqrt{u}.

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 15x^2.

Now, using the chain rule, we have:

\frac{dh}{dx} = \frac{1}{2\sqrt{u}} \cdot 15x^2

\frac{dh}{dx} = \frac{15x^2}{2\sqrt{5x^3 + 2}}.

**4. Determine the derivative of k(x) = \cos(2x^2 + 3).**

**Solution**:

To differentiate a function of a function, we can use the chain rule. Let u = 2x^2 + 3, then k(x) = \cos(u).

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 4x.

Now, applying the chain rule, we get:

\frac{dk}{dx} = -\sin(u) \cdot 4x

\frac{dk}{dx} = -4x\sin(2x^2 + 3).

**5. Find the derivative of m(x) = e^{x^2 + 2x + 3}.**

**Solution**:

To differentiate a function of a function, we can use the chain rule. Let u = x^2 + 2x + 3, then m(x) = e^u.

Now, let's differentiate u with respect to x:

\frac{du}{dx} = 2x + 2.

Now, applying the chain rule, we have:

\frac{dm}{dx} = e^u \cdot (2x + 2)

\frac{dm}{dx} = (2x + 2)e^{x^2 + 2x + 3}.

**Questions of Functions of Function**

1. Find the derivative of the function f(x) = \sin(3x^2).

2. Calculate the derivative of g(x) = e^{2x^3}.

3. Determine the derivative of h(x) = \cos(4x).

4. Find the derivative of p(x) = \ln(5x^2 + 3).

5. Calculate the derivative of q(x) = \sqrt{2x + 1}.

6. Determine the derivative of r(x) = \tan(2x^2).

7. Find the derivative of s(x) = e^{\frac{1}{x^2}}.

8. Calculate the derivative of t(x) = \ln(4x^3 - 2x + 5).

9. Determine the derivative of u(x) = \sin(5x).

10. Find the derivative of v(x) = \cos(3x^2 + 2x).

11. Calculate the derivative of w(x) = e^{6x^2 + 3x}.

12. Determine the derivative of y(x) = \ln(2x+1)^2.

13. Find the derivative of z(x) = \sqrt{3x^2 + 1}.

14. Calculate the derivative of a(x) = \tan(4x^3 + x).

15. Determine the derivative of b(x) = e^{8x^2 - 2x}.

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**Corresponding Answers of Functions of Function**

1. The derivative of f(x) = \sin(3x^2) is f'(x) = 6x\cos(3x^2).

2. The derivative of g(x) = e^{2x^3} is g'(x) = 6x^2e^{2x^3}.

3. The derivative of h(x) = \cos(4x) is h'(x) = -4\sin(4x).

4. The derivative of p(x) = \ln(5x^2 + 3) is p'(x) = \frac{10x}{5x^2 + 3}.

5. The derivative of q(x) = \sqrt{2x + 1} is q'(x) = \frac{1}{\sqrt{2x + 1}}.

6. The derivative of r(x) = \tan(2x^2) is r'(x) = 4x\sec^2(2x^2).

7. The derivative of s(x) = e^{\frac{1}{x^2}} is s'(x) = -\frac{2}{x^3}e^{\frac{1}{x^2}}.

8. The derivative of t(x) = \ln(4x^3 - 2x + 5) is t'(x) = \frac{12x^2 - 2}{4x^3 - 2x + 5}.

9. The derivative of u(x) = \sin(5x) is u'(x) = 5\cos(5x).

10. The derivative of v(x) = \cos(3x^2 + 2x) is v'(x) = -6x\sin(3x^2 + 2x) - 2\sin(3x^2 + 2x).

11. The derivative of w(x) = e^{6x^2 + 3x} is w'(x) = (12x + 3)e^{6x^2 + 3x}.

12. The derivative of y(x) = \ln(2x+1)^2 is y'(x) = \frac{2}{2x+1}.

13. The derivative of z(x) = \sqrt{3x^2 + 1} is z'(x) = \frac{3x}{\sqrt{3x^2 + 1}}.

14. The derivative of a(x) = \tan(4x^3 + x) is a'(x) = 12x^2\sec^2(4x^3 + x) + \sec^2(4x^3 + x).

15. The derivative of b(x) = e^{8x^2 - 2x} is b'(x) = (16x - 2)e^{8x^2 - 2x}.

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