I see that you've uploaded a worksheet with several calculus problems, specifically focusing on finding derivatives of various functions. I'll go through each question and provide a solution.

Problem 1: Calculate the Derivative of Each Function Using Derivative Properties

1. (a) $f(x) = x^{100}$

   Using the power rule $f'(x) = n \cdot x^{n-1}$: $f'(x) = 100 \cdot x^{99}$

2. (b) $f(x) = 5x^2 + 7x - 6$

   Differentiate term-by-term: $f'(x) = 2 \cdot 5x^{2-1} + 7 \cdot 1 - 0 = 10x + 7$

3. (c) $f(x) = \frac{3x - 5}{x^2 + 7}$

   Use the quotient rule $\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}$:

   • Let $u = 3x - 5$ and $v = x^2 + 7$.
   • Then $u' = 3$ and $v' = 2x$.

   $f'(x) = \frac{(3)(x^2 + 7) - (3x - 5)(2x)}{(x^2 + 7)^2}$ Simplify further as needed.

Problem 2: Calculate the Derivative and Evaluate at Specific Points

1. (a) $f(x) = 9x^3 + 2x^2 + 6x + 6$, find $f'(3)$

   Differentiate term-by-term: $f'(x) = 27x^2 + 4x + 6$ Substitute $x = 3$: $f'(3) = 27(3)^2 + 4(3) + 6 = 243 + 12 + 6 = 261$

2. (b) $f(x) = (2x^3 - 3x^2)(5x^2 - 7)$, find $f'(2)$

   Use the product rule $(uv)' = u'v + uv'$:

   • Let $u = 2x^3 - 3x^2$ and $v = 5x^2 - 7$.
   • Then $u' = 6x^2 - 6x$ and $v' = 10x$.

   $f'(x) = (6x^2 - 6x)(5x^2 - 7) + (2x^3 - 3x^2)(10x)$ Substitute $x = 2$ and simplify.

3. (c) $f(x) = \frac{6}{2x - 5} - \frac{x}{x^2}$, find $f'(1)$

   Simplify first if possible, then use the quotient rule as needed. Substitute $x = 1$ after finding $f'(x)$.

Problem 3: Given $y = (2x^3 - x^2 + 1)^3$, Find $y'$

Using the chain rule, let $u = 2x^3 - x^2 + 1$, so $y = u^3$.

• Then $y' = 3u^2 \cdot u'$.
• $u' = 6x^2 - 2x$, so $y' = 3(2x^3 - x^2 + 1)^2 \cdot (6x^2 - 2x)$

Problem 4: Find the First Derivative of Each Trigonometric Function

1. (a) $y = \frac{1 - \cos^2 x}{\sin x + \cos x}$

   Simplify using trigonometric identities first, then differentiate using the quotient rule.

2. (b) $y = \frac{\cos x}{\sin x + \cos x}$

   Use the quotient rule to find $y'$.

Problem 5: First Derivative of $f(x) = 3x^2 \sin(2x) + \cos^2(x)$

1. Differentiate each term separately, using the product rule for $3x^2 \sin(2x)$ and chain rule for $\cos^2(x)$.

Would you like detailed steps for each solution, or do you have specific questions?

Follow-up Questions

1. How does the quotient rule apply to different functions?
2. What are the common mistakes to avoid when using the product rule?
3. Why is it useful to simplify expressions before differentiating?
4. What are some typical applications of the chain rule in calculus?
5. How do trigonometric identities help in simplifying derivatives?

Tip

When solving complex derivatives, try to simplify the function first if possible; it often makes the differentiation process much easier.