Question Statement

Find the first and second derivatives of the function:

$f (θ) = sin^{2} 5 θ$

Background and Explanation

This problem requires applying the chain rule for composite functions, where we differentiate from the outermost function inward. You'll also need the double angle identity for sine to simplify the first derivative result.

Solution

Finding the First Derivative $f^{'} (θ)$

Given the function: $f (θ) = sin^{2} 5 θ$

We differentiate with respect to $θ$ using the chain rule. Treat this as a composition of functions: outer function is $(\cdot)^{2}$ and inner function is $sin 5 θ$ .

$f^{'} (θ) = 2 sin^{2 - 1} 5 θ \cdot \frac{d}{d θ} (sin 5 θ) = 2 sin 5 θ \cdot (cos 5 θ) \cdot \frac{d}{d θ} (5 θ) = 2 sin 5 θ \cdot cos 5 θ \cdot 5 = 10 sin 5 θ cos 5 θ$

Using the double angle identity $sin 2 x = 2 sin x cos x$ , we can simplify: $2 sin 5 θ cos 5 θ = sin (2 \cdot 5 θ) = sin 10 θ$

Therefore: $f^{'} (θ) = 5 sin 10 θ$

Finding the Second Derivative $f^{''} (θ)$

Differentiating $f^{'} (θ) = 5 sin 10 θ$ with respect to $θ$ :

$f^{''} (θ) = 5 cos (10 θ) \cdot \frac{d}{d θ} (10 θ) = 5 cos 10 θ \cdot 10 = 50 cos 10 θ$

Key Formulas or Methods Used

Chain Rule: $\frac{d}{d θ} [f (g (θ))] = f^{'} (g (θ)) \cdot g^{'} (θ)$
Power Rule: $\frac{d}{d θ} [u^{n}] = n u^{n - 1} \frac{d u}{d θ}$
Derivative of Sine: $\frac{d}{d θ} [sin u] = cos u \cdot \frac{d u}{d θ}$
Double Angle Identity: $sin 2 x = 2 sin x cos x$ (used to simplify $2 sin 5 θ cos 5 θ$ to $sin 10 θ$ )

Summary of Steps

Apply chain rule to $sin^{2} 5 θ$ : differentiate the outer square function, then multiply by derivative of $sin 5 θ$ , then multiply by derivative of $5 θ$
Simplify the result $10 sin 5 θ cos 5 θ$ using the double angle identity to get $5 sin 10 θ$
Differentiate again using chain rule on $5 sin 10 θ$ to obtain the second derivative $50 cos 10 θ$

Question Statement

Find the first and second derivatives of the function:

$f (θ) = sin^{2} 5 θ$

Background and Explanation

Solution

Finding the First Derivative $f^{'} (θ)$

Given the function: $f (θ) = sin^{2} 5 θ$

We differentiate with respect to $θ$ using the chain rule. Treat this as a composition of functions: outer function is $(\cdot)^{2}$ and inner function is $sin 5 θ$ .

$f^{'} (θ) = 2 sin^{2 - 1} 5 θ \cdot \frac{d}{d θ} (sin 5 θ) = 2 sin 5 θ \cdot (cos 5 θ) \cdot \frac{d}{d θ} (5 θ) = 2 sin 5 θ \cdot cos 5 θ \cdot 5 = 10 sin 5 θ cos 5 θ$

Using the double angle identity $sin 2 x = 2 sin x cos x$ , we can simplify: $2 sin 5 θ cos 5 θ = sin (2 \cdot 5 θ) = sin 10 θ$

Therefore: $f^{'} (θ) = 5 sin 10 θ$

Finding the Second Derivative $f^{''} (θ)$

Differentiating $f^{'} (θ) = 5 sin 10 θ$ with respect to $θ$ :

$f^{''} (θ) = 5 cos (10 θ) \cdot \frac{d}{d θ} (10 θ) = 5 cos 10 θ \cdot 10 = 50 cos 10 θ$

Key Formulas or Methods Used

Chain Rule: $\frac{d}{d θ} [f (g (θ))] = f^{'} (g (θ)) \cdot g^{'} (θ)$
Power Rule: $\frac{d}{d θ} [u^{n}] = n u^{n - 1} \frac{d u}{d θ}$
Derivative of Sine: $\frac{d}{d θ} [sin u] = cos u \cdot \frac{d u}{d θ}$
Double Angle Identity: $sin 2 x = 2 sin x cos x$ (used to simplify $2 sin 5 θ cos 5 θ$ to $sin 10 θ$ )

Summary of Steps

Apply chain rule to $sin^{2} 5 θ$ : differentiate the outer square function, then multiply by derivative of $sin 5 θ$ , then multiply by derivative of $5 θ$
Simplify the result $10 sin 5 θ cos 5 θ$ using the double angle identity to get $5 sin 10 θ$
Differentiate again using chain rule on $5 sin 10 θ$ to obtain the second derivative $50 cos 10 θ$

2.8 Q-11

Question Statement

Background and Explanation

Solution

Finding the First Derivative $f^{'} (θ)$

Finding the Second Derivative $f^{''} (θ)$

Key Formulas or Methods Used

Summary of Steps

2.8 Q-11

Question Statement

Background and Explanation

Solution

Finding the First Derivative $f^{'} (θ)$

Finding the Second Derivative $f^{''} (θ)$

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Finding the First Derivative f′(θ)

Finding the Second Derivative f′′(θ)

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Finding the First Derivative f′(θ)

Finding the Second Derivative f′′(θ)

Key Formulas or Methods Used

Summary of Steps

Finding the First Derivative $f^{'} (θ)$

Finding the Second Derivative $f^{''} (θ)$

Finding the First Derivative $f^{'} (θ)$

Finding the Second Derivative $f^{''} (θ)$