Question Statement

Find the first and second derivatives of the function $y = - x^{3} + 6 x + 9$ with respect to $x$ .

Background and Explanation

This problem requires applying the power rule for differentiation, which states that $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ , along with the constant multiple rule and sum rule. These fundamental rules allow us to find the rate of change (first derivative) and the rate of change of the rate of change (second derivative) for polynomial functions.

Solution

Finding the First Derivative $\frac{d y}{d x}$

We begin with the function: $y = - x^{3} + 6 x + 9$

To find the first derivative, we differentiate each term with respect to $x$ using the power rule. The derivative of a sum is the sum of the derivatives, and constants can be factored out:

$\frac{d y}{d x} = - \frac{d}{d x} (x^{3}) + 6 \frac{d}{d x} (x) + \frac{d}{d x} (9)$

Applying the power rule $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ to each term:

For $x^{3}$ : the derivative is $3 x^{2}$ , so $- \frac{d}{d x} (x^{3}) = - 3 x^{2}$
For $x$ : the derivative is $1$ , so $6 \frac{d}{d x} (x) = 6 (1) = 6$
For the constant $9$ : the derivative is $0$

Therefore: $\frac{d y}{d x} = - 3 x^{2} + 6 + 0$

$\frac{d y}{d x} = - 3 x^{2} + 6$

Finding the Second Derivative $\frac{d ^{2} y}{d x ^{2}}$

To find the second derivative, we differentiate the first derivative with respect to $x$ :

$\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- 3 x^{2} + 6)$

Breaking this down term by term: $\frac{d ^{2} y}{d x ^{2}} = - 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (6)$

Applying the power rule to $x^{2}$ (derivative is $2 x$ ) and the constant rule to $6$ (derivative is $0$ ): $\frac{d ^{2} y}{d x ^{2}} = - 3 (2 x) + 0$

$\frac{d ^{2} y}{d x ^{2}} = - 6 x$

Key Formulas or Methods Used

Power Rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$
Constant Rule: $\frac{d}{d x} (c) = 0$ where $c$ is a constant
Constant Multiple Rule: $\frac{d}{d x} [c \cdot f (x)] = c \cdot \frac{d}{d x} [f (x)]$
Sum Rule: $\frac{d}{d x} [f (x) + g (x)] = \frac{d}{d x} [f (x)] + \frac{d}{d x} [g (x)]$

Summary of Steps

Identify the function: $y = - x^{3} + 6 x + 9$
Apply the power rule to each term to find the first derivative: $\frac{d y}{d x} = - 3 x^{2} + 6$
Differentiate the result again to find the second derivative: $\frac{d ^{2} y}{d x ^{2}} = - 6 x$

Background and Explanation

This problem requires applying the power rule for differentiation, which states that

\frac{d}{d x} (x^{n}) = n x^{n - 1}

, along with the constant multiple rule and sum rule. These fundamental rules allow us to find the rate of change (first derivative) and the rate of change of the rate of change (second derivative) for polynomial functions.

Finding the First Derivative

\frac{d y}{d x}

We begin with the function:

y = - x^{3} + 6 x + 9

To find the first derivative, we differentiate each term with respect to

x

using the power rule. The derivative of a sum is the sum of the derivatives, and constants can be factored out:

\frac{d y}{d x} = - \frac{d}{d x} (x^{3}) + 6 \frac{d}{d x} (x) + \frac{d}{d x} (9)

Applying the power rule

\frac{d}{d x} (x^{n}) = n x^{n - 1}

to each term:

For

x^{3}

: the derivative is

3 x^{2}

, so

- \frac{d}{d x} (x^{3}) = - 3 x^{2}

For

x

: the derivative is

1

, so

6 \frac{d}{d x} (x) = 6 (1) = 6

For the constant

9

: the derivative is

0

Therefore:

\frac{d y}{d x} = - 3 x^{2} + 6 + 0

\frac{d y}{d x} = - 3 x^{2} + 6

Finding the Second Derivative

\frac{d ^{2} y}{d x ^{2}}

To find the second derivative, we differentiate the first derivative with respect to

x

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- 3 x^{2} + 6)

Breaking this down term by term:

\frac{d ^{2} y}{d x ^{2}} = - 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (6)

Applying the power rule to

x^{2}

(derivative is

2 x

) and the constant rule to

6

(derivative is

0

\frac{d ^{2} y}{d x ^{2}} = - 3 (2 x) + 0

\frac{d ^{2} y}{d x ^{2}} = - 6 x

2.8 Q-1

Question Statement

Background and Explanation

Solution

Finding the First Derivative $\frac{d y}{d x}$

Finding the Second Derivative $\frac{d ^{2} y}{d x ^{2}}$

Key Formulas or Methods Used

Summary of Steps

2.8 Q-1

Question Statement

Background and Explanation

Solution

Finding the First Derivative $\frac{d y}{d x}$

Finding the Second Derivative $\frac{d ^{2} y}{d x ^{2}}$

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Finding the First Derivative dxdy​

Finding the Second Derivative dx2d2y​

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Finding the First Derivative dxdy​

Finding the Second Derivative dx2d2y​

Key Formulas or Methods Used

Summary of Steps

Finding the First Derivative $\frac{d y}{d x}$

Finding the Second Derivative $\frac{d ^{2} y}{d x ^{2}}$

Finding the First Derivative $\frac{d y}{d x}$

Finding the Second Derivative $\frac{d ^{2} y}{d x ^{2}}$