Question Statement

Given the function $f (x) = cos (10 x)$ , find the first derivative $f^{'} (x)$ and the second derivative $f^{''} (x)$ .

Background and Explanation

This problem involves differentiating composite trigonometric functions using the chain rule, which states that the derivative of $f (g (x))$ equals $f^{'} (g (x)) \cdot g^{'} (x)$ . You should be familiar with the basic derivatives of sine and cosine functions, as well as how to handle constant coefficients inside the argument.

Solution

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

We start with the given function: $f (x) = cos (10 x)$

To find $f^{'} (x)$ , we differentiate with respect to $x$ using the chain rule. The outer function is $cos (u)$ and the inner function is $u = 10 x$ .

$f^{'} (x) f^{'} (x) = - sin (10 x) \cdot \frac{d}{d x} (10 x) = - sin (10 x) \cdot (10) = - 10 sin (10 x)$

The negative sign comes from the derivative of cosine, and we multiply by the derivative of the inner function $10 x$ , which is $10$ .

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

Now we differentiate $f^{'} (x)$ with respect to $x$ again to find the second derivative:

$f^{'} (x) = - 10 sin (10 x)$

Applying the chain rule once more (where the derivative of $sin (u)$ is $cos (u) \cdot u^{'}$ ):

$f^{''} (x) = - 10 \cdot \frac{d}{d x} (sin (10 x)) = - 10 \cdot (cos (10 x)) \cdot 10 = - 100 cos (10 x)$

Here, the constant $- 10$ remains as a coefficient, and we differentiate $sin (10 x)$ to get $cos (10 x) \cdot 10$ .

Key Formulas or Methods Used

Chain Rule: $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$
Derivative of Cosine: $\frac{d}{d x} (cos u) = - sin (u) \cdot \frac{d u}{d x}$
Derivative of Sine: $\frac{d}{d x} (sin u) = cos (u) \cdot \frac{d u}{d x}$
Power Rule for Linear Terms: $\frac{d}{d x} (a x) = a$

Summary of Steps

Identify the composite structure: Recognize $f (x) = cos (10 x)$ has outer function $cos (u)$ and inner function $u = 10 x$ .
Apply chain rule for $f^{'} (x)$ : Differentiate the outer function ( $- sin (10 x)$ ) and multiply by the derivative of the inner function ( $10$ ).
Simplify: Obtain $f^{'} (x) = - 10 sin (10 x)$ .
Differentiate again for $f^{''} (x)$ : Apply the chain rule to $- 10 sin (10 x)$ , keeping the constant coefficient $- 10$ .
Apply chain rule: Differentiate $sin (10 x)$ to get $cos (10 x) \cdot 10$ , then multiply by $- 10$ .
Final simplification: Obtain $f^{''} (x) = - 100 cos (10 x)$ .

Background and Explanation

This problem involves differentiating composite trigonometric functions using the chain rule, which states that the derivative of

f (g (x))

equals

f^{'} (g (x)) \cdot g^{'} (x)

. You should be familiar with the basic derivatives of sine and cosine functions, as well as how to handle constant coefficients inside the argument.

Finding the First Derivative

f^{'} (x)

We start with the given function:

f (x) = cos (10 x)

To find

f^{'} (x)

, we differentiate with respect to

x

using the chain rule. The outer function is

cos (u)

and the inner function is

u = 10 x

f^{'} (x) f^{'} (x) = - sin (10 x) \cdot \frac{d}{d x} (10 x) = - sin (10 x) \cdot (10) = - 10 sin (10 x)

The negative sign comes from the derivative of cosine, and we multiply by the derivative of the inner function

10 x

, which is

10

Finding the Second Derivative

f^{''} (x)

Now we differentiate

f^{'} (x)

with respect to

x

again to find the second derivative:

f^{'} (x) = - 10 sin (10 x)

Applying the chain rule once more (where the derivative of

sin (u)

cos (u) \cdot u^{'}

f^{''} (x) = - 10 \cdot \frac{d}{d x} (sin (10 x)) = - 10 \cdot (cos (10 x)) \cdot 10 = - 100 cos (10 x)

Here, the constant

- 10

remains as a coefficient, and we differentiate

sin (10 x)

to get

cos (10 x) \cdot 10

Summary of Steps

Identify the composite structure: Recognize

f (x) = cos (10 x)

has outer function

cos (u)

and inner function

u = 10 x

Apply chain rule for $f^{'} (x)$ : Differentiate the outer function (

- sin (10 x)

) and multiply by the derivative of the inner function (

10

Simplify: Obtain

f^{'} (x) = - 10 sin (10 x)

Differentiate again for $f^{''} (x)$ : Apply the chain rule to

- 10 sin (10 x)

, keeping the constant coefficient

- 10

Apply chain rule: Differentiate

sin (10 x)

to get

cos (10 x) \cdot 10

, then multiply by

- 10

Final simplification: Obtain

f^{''} (x) = - 100 cos (10 x)

2.8 Q-9

Question Statement

Background and Explanation

Solution

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

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

Key Formulas or Methods Used

Summary of Steps

2.8 Q-9

Question Statement

Background and Explanation

Solution

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

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

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Finding the First Derivative f′(x)

Finding the Second Derivative f′′(x)

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Finding the First Derivative f′(x)

Finding the Second Derivative f′′(x)

Key Formulas or Methods Used

Summary of Steps

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

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

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

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