Let's go through each question in the image step-by-step.

Question 1

Given: $h(x) = \sqrt{x} + \frac{3}{x}$ Find $h'(x)$.

Solution:

To differentiate, we need to find the derivatives of each term separately.

1. Rewrite $h(x)$: $h(x) = x^{1/2} + 3x^{-1}$

2. Differentiate each term:

   • The derivative of $x^{1/2}$ is $\frac{1}{2}x^{-1/2}$.
   • The derivative of $3x^{-1}$ is $-3x^{-2}$.

3. Combine: $h'(x) = \frac{1}{2}x^{-1/2} - 3x^{-2}$ Which can also be written as: $h'(x) = \frac{1}{2\sqrt{x}} - \frac{3}{x^2}$

Question 2

Given: $f(x) = \frac{x + 2}{x^3 - 1}$ Find $f'(x)$.

Solution:

This is a quotient, so we use the quotient rule: if $f(x) = \frac{u}{v}$, then $f'(x) = \frac{u'v - uv'}{v^2}$.

1. Define $u$ and $v$:

   • $u = x + 2$ and $u' = 1$
   • $v = x^3 - 1$ and $v' = 3x^2$

2. Apply the quotient rule: $f'(x) = \frac{(1)(x^3 - 1) - (x + 2)(3x^2)}{(x^3 - 1)^2}$

3. Simplify: $f'(x) = \frac{x^3 - 1 - 3x^3 - 6x^2}{(x^3 - 1)^2} = \frac{-2x^3 - 6x^2 - 1}{(x^3 - 1)^2}$

Question 3

Given $f(x) = 5x^3 + 2x^2 + 6x + 12$, find $f'(x)$.

Solution:

Differentiate term-by-term.

1. The derivative of $5x^3$ is $15x^2$.
2. The derivative of $2x^2$ is $4x$.
3. The derivative of $6x$ is $6$.
4. The derivative of $12$ is $0$.

Combine these to get: $f'(x) = 15x^2 + 4x + 6$

Question 4

Given $f(x) = 3x^4 + 2x^3 - x + 2$, find $f'(1)$.

Solution:

1. Differentiate term-by-term:

   • The derivative of $3x^4$ is $12x^3$.
   • The derivative of $2x^3$ is $6x^2$.
   • The derivative of $-x$ is $-1$.
   • The derivative of $2$ is $0$.

   So, $f'(x) = 12x^3 + 6x^2 - 1$

2. Substitute $x = 1$ to find $f'(1)$: $f'(1) = 12(1)^3 + 6(1)^2 - 1 = 12 + 6 - 1 = 17$

Summary of Answers:

1. $h'(x) = \frac{1}{2\sqrt{x}} - \frac{3}{x^2}$
2. $f'(x) = \frac{-2x^3 - 6x^2 - 1}{(x^3 - 1)^2}$
3. $f'(x) = 15x^2 + 4x + 6$
4. $f'(1) = 17$

Would you like further details or have any questions?

Related Questions

1. How can the chain rule be applied to composite functions in differentiation?
2. How does the product rule differ from the quotient rule, and when should each be used?
3. What is the geometric interpretation of a derivative at a point on a curve?
4. How can higher-order derivatives provide information about the shape of a graph?
5. How do you determine points of inflection using the second derivative?

Tip

Remember that simplifying expressions after applying differentiation rules can help in verifying the correctness of your answer.