1. Differentiate. *h*(𝜃) = 𝜃2 sin(𝜃). *h* ′(𝜃) =       2.Differentiate. *y* = sec(𝜃) tan(𝜃)*. y'* =   3.Differentiate. *g*(*t*) = 8 sec *t* + 9 tan *t.  g'*(*t*) =  4.Differentiate. *y* = (8*x)*/(9 − tan(*x*)) *y* ′ =

Let's differentiate the functions one by one.

### 1. Differentiate \( h(\theta) = \theta^2 \sin(\theta) \)

To differentiate this, we use the **product rule**. Recall that if \( h(\theta) = u(\theta) v(\theta) \), then

\[
h'(\theta) = u'(\theta) v(\theta) + u(\theta) v'(\theta)
\]

Let:
- \( u(\theta) = \theta^2 \)
- \( v(\theta) = \sin(\theta) \)

Now differentiate each part:
- \( u'(\theta) = 2\theta \)
- \( v'(\theta) = \cos(\theta) \)

Using the product rule:

\[
h'(\theta) = (2\theta) \sin(\theta) + \theta^2 \cos(\theta)
\]

Thus:

\[
h'(\theta) = 2\theta \sin(\theta) + \theta^2 \cos(\theta)
\]

---

### 2. Differentiate \( y = \sec(\theta) \tan(\theta) \)

We use the **product rule** again. Let:
- \( u(\theta) = \sec(\theta) \)
- \( v(\theta) = \tan(\theta) \)

Now differentiate each part:
- \( u'(\theta) = \sec(\theta) \tan(\theta) \)
- \( v'(\theta) = \sec^2(\theta) \)

Using the product rule:

\[
y' = (\sec(\theta) \tan(\theta)) \tan(\theta) + \sec(\theta) \sec^2(\theta)
\]

Thus:

\[
y' = \sec(\theta) \tan^2(\theta) + \sec^3(\theta)
\]

---

### 3. Differentiate \( g(t) = 8 \sec(t) + 9 \tan(t) \)

For this function, simply apply the basic differentiation rules for the secant and tangent functions.

- \( \frac{d}{dt} \sec(t) = \sec(t) \tan(t) \)
- \( \frac{d}{dt} \tan(t) = \sec^2(t) \)

Thus:

\[
g'(t) = 8 \sec(t) \tan(t) + 9 \sec^2(t)
\]

---

### 4. Differentiate \( y = \frac{8x}{9 - \tan(x)} \)

We use the **quotient rule** here. Recall that if \( y = \frac{u(x)}{v(x)} \), then

\[
y' = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}
\]

Let:
- \( u(x) = 8x \)
- \( v(x) = 9 - \tan(x) \)

Now differentiate each part:
- \( u'(x) = 8 \)
- \( v'(x) = -\sec^2(x) \)

Using the quotient rule:

\[
y' = \frac{(8)(9 - \tan(x)) - (8x)(-\sec^2(x))}{(9 - \tan(x))^2}
\]

Simplifying:

\[
y' = \frac{72 - 8\tan(x) + 8x\sec^2(x)}{(9 - \tan(x))^2}
\]

---

Let me know if you want details or have any questions!

Here are some related questions:
1. What is the product rule for differentiation, and how is it applied?
2. Can you explain how to use the quotient rule for differentiation?
3. How do trigonometric identities affect differentiation results?
4. What is the significance of higher-order derivatives for trigonometric functions?
5. How do we simplify complex derivatives involving trigonometric functions?

**Tip:** Always simplify expressions before differentiating to avoid errors in applying rules.

1. Differentiate. h(𝜃) = 𝜃^2 sin(𝜃). h ′(𝜃) =  2. Differentiate. y = sec(𝜃) tan(𝜃). y' =  3. Differentiate. g(t) = 8 sec t + 9 tan t. g' (t) =  4. Differentiate. y = (8*x)/(9 − tan(x)) y'  = 

This guide explains how to differentiate functions involving trigonometric terms such as h(𝜃) = 𝜃^2 sin(𝜃) and y = sec(𝜃) tan(𝜃), using key calculus concepts like the product rule and quotient rule. Step-by-step solutions are provided for each function. Ideal for students in Grades 11-12 learning advanced calculus.

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