Question Statement

Given the function:

$y = 2 x^{6} + 5 x^{3} - 6 x^{2}$

Find the first derivative $\frac{d y}{d x}$ and the second derivative $\frac{d ^{2} y}{d x ^{2}}$ .

Background and Explanation

This problem requires applying the power rule for differentiation, which states that the derivative of $x^{n}$ is $n x^{n - 1}$ . Since this is a polynomial, we differentiate term by term while preserving the constant coefficients.

Solution

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

We begin with the given function:

$y = 2 x^{6} + 5 x^{3} - 6 x^{2}$

To find $\frac{d y}{d x}$ , we differentiate each term with respect to $x$ using the power rule and constant multiple rule. We treat each term separately:

$\frac{d y}{d x} = 2 \frac{d}{d x} (x^{6}) + 5 \frac{d}{d x} (x^{3}) - 6 \frac{d}{d x} (x^{2}) = 2 (6 x^{5}) + 5 (3 x^{2}) - 6 (2 x) = 12 x^{5} + 15 x^{2} - 12 x$

Reasoning: For each term, we multiply the coefficient by the exponent, then reduce the exponent by 1. The derivative of $x^{6}$ is $6 x^{5}$ , multiplied by 2 gives $12 x^{5}$ . Similarly, $5 \times 3 x^{2} = 15 x^{2}$ and $- 6 \times 2 x = - 12 x$ .

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

Now we differentiate $\frac{d y}{d x}$ with respect to $x$ again to obtain the second derivative:

$\frac{d ^{2} y}{d x ^{2}} = 12 \frac{d}{d x} (x^{5}) + 15 \frac{d}{d x} (x^{2}) - 12 \frac{d}{d x} (x) = 12 (5 x^{4}) + 15 (2 x) - 12 (1) = 60 x^{4} + 30 x - 12$

Reasoning: We apply the power rule once more to each term of the first derivative. Note that the derivative of $x$ (which is $x^{1}$ ) is 1, so the last term becomes $- 12 (1) = - 12$ .

Key Formulas or Methods Used

Power Rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$
Constant Multiple Rule: $\frac{d}{d x} [c \cdot f (x)] = c \cdot \frac{d}{d x} [f (x)]$
Sum/Difference Rule: $\frac{d}{d x} [f (x) \pm g (x)] = \frac{d}{d x} [f (x)] \pm \frac{d}{d x} [g (x)]$

Summary of Steps

Identify the function: $y = 2 x^{6} + 5 x^{3} - 6 x^{2}$
Apply the power rule to each term to find the first derivative: multiply coefficients by exponents and reduce exponents by 1
Simplify to obtain $\frac{d y}{d x} = 12 x^{5} + 15 x^{2} - 12 x$
Differentiate the first derivative again using the power rule to find the second derivative
Simplify to obtain $\frac{d ^{2} y}{d x ^{2}} = 60 x^{4} + 30 x - 12$

Finding the First Derivative

\frac{d y}{d x}

We begin with the given function:

y = 2 x^{6} + 5 x^{3} - 6 x^{2}

To find

\frac{d y}{d x}

, we differentiate each term with respect to

x

using the power rule and constant multiple rule. We treat each term separately:

\frac{d y}{d x} = 2 \frac{d}{d x} (x^{6}) + 5 \frac{d}{d x} (x^{3}) - 6 \frac{d}{d x} (x^{2}) = 2 (6 x^{5}) + 5 (3 x^{2}) - 6 (2 x) = 12 x^{5} + 15 x^{2} - 12 x

Reasoning: For each term, we multiply the coefficient by the exponent, then reduce the exponent by 1. The derivative of

x^{6}

6 x^{5}

, multiplied by 2 gives

12 x^{5}

. Similarly,

5 \times 3 x^{2} = 15 x^{2}

and

- 6 \times 2 x = - 12 x

Finding the Second Derivative

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

Now we differentiate

\frac{d y}{d x}

with respect to

x

again to obtain the second derivative:

\frac{d ^{2} y}{d x ^{2}} = 12 \frac{d}{d x} (x^{5}) + 15 \frac{d}{d x} (x^{2}) - 12 \frac{d}{d x} (x) = 12 (5 x^{4}) + 15 (2 x) - 12 (1) = 60 x^{4} + 30 x - 12

Reasoning: We apply the power rule once more to each term of the first derivative. Note that the derivative of

x

(which is

x^{1}

) is 1, so the last term becomes

- 12 (1) = - 12

Summary of Steps

Identify the function:

y = 2 x^{6} + 5 x^{3} - 6 x^{2}

Apply the power rule to each term to find the first derivative: multiply coefficients by exponents and reduce exponents by 1

Simplify to obtain

\frac{d y}{d x} = 12 x^{5} + 15 x^{2} - 12 x

Differentiate the first derivative again using the power rule to find the second derivative

Simplify to obtain

\frac{d ^{2} y}{d x ^{2}} = 60 x^{4} + 30 x - 12

2.8 Q-4

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-4

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