Math / Calculus

QuestionQuestion Watch Video Show Examples Given the function $y=\frac{1}{x^{3}}$ , find $\frac{d y}{d x}$ . Express your answer in simplest form without using negative exponents.

Studdy Solution

STEP 1

1. We are given the function $y = \frac{1}{x^3}$ .
2. We need to find the derivative $\frac{dy}{dx}$ .
3. The answer should be expressed in simplest form without using negative exponents.

STEP 2

1. Rewrite the function using negative exponents.
2. Differentiate the function using the power rule.
3. Simplify the derivative and express it without negative exponents.

STEP 3

Rewrite the function using negative exponents:
$y = x^{-3}$

STEP 4

Differentiate the function using the power rule, which states that if $y = x^n$ , then $\frac{dy}{dx} = nx^{n-1}$ :
$\frac{dy}{dx} = -3x^{-3-1}$ $\frac{dy}{dx} = -3x^{-4}$

STEP 5

Simplify the derivative and express it without negative exponents:
$\frac{dy}{dx} = -\frac{3}{x^4}$
The derivative of the function is:
$\boxed{-\frac{3}{x^4}}$

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QuestionQuestion Watch Video Show Examples Given the function $y=\frac{1}{x^{3}}$ , find $\frac{d y}{d x}$ . Express your answer in simplest form without using negative exponents.

Studdy Solution

STEP 1

1. We are given the function $y = \frac{1}{x^3}$ .
2. We need to find the derivative $\frac{dy}{dx}$ .
3. The answer should be expressed in simplest form without using negative exponents.

STEP 2

1. Rewrite the function using negative exponents.
2. Differentiate the function using the power rule.
3. Simplify the derivative and express it without negative exponents.

STEP 3

Rewrite the function using negative exponents:
$y = x^{-3}$

STEP 4

Differentiate the function using the power rule, which states that if $y = x^n$ , then $\frac{dy}{dx} = nx^{n-1}$ :
$\frac{dy}{dx} = -3x^{-3-1}$ $\frac{dy}{dx} = -3x^{-4}$

STEP 5

Simplify the derivative and express it without negative exponents:
$\frac{dy}{dx} = -\frac{3}{x^4}$
The derivative of the function is:
$\boxed{-\frac{3}{x^4}}$

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QuestionQuestion Watch Video Show Examples Given the function y=1x3y=\frac{1}{x^{3}}y=x31​, find dydx\frac{d y}{d x}dxdy​. Express your answer in simplest form without using negative exponents.

Studdy Solution

STEP 1

1. We are given the function y=1x3 y = \frac{1}{x^3} y=x31​.2. We need to find the derivative dydx\frac{dy}{dx}dxdy​.3. The answer should be expressed in simplest form without using negative exponents.

STEP 2

1. Rewrite the function using negative exponents.2. Differentiate the function using the power rule.3. Simplify the derivative and express it without negative exponents.

STEP 3

Rewrite the function using negative exponents:y=x−3 y = x^{-3} y=x−3

STEP 4

Differentiate the function using the power rule, which states that if y=xn y = x^n y=xn, then dydx=nxn−1\frac{dy}{dx} = nx^{n-1}dxdy​=nxn−1:dydx=−3x−3−1 \frac{dy}{dx} = -3x^{-3-1} dxdy​=−3x−3−1 dydx=−3x−4 \frac{dy}{dx} = -3x^{-4} dxdy​=−3x−4

STEP 5

Simplify the derivative and express it without negative exponents:dydx=−3x4 \frac{dy}{dx} = -\frac{3}{x^4} dxdy​=−x43​ The derivative of the function is:−3x4 \boxed{-\frac{3}{x^4}} −x43​​

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QuestionQuestion Watch Video Show Examples Given the function $y=\frac{1}{x^{3}}$ , find $\frac{d y}{d x}$ . Express your answer in simplest form without using negative exponents.

1. We are given the function $y = \frac{1}{x^3}$ .
2. We need to find the derivative $\frac{dy}{dx}$ .
3. The answer should be expressed in simplest form without using negative exponents.

1. Rewrite the function using negative exponents.
2. Differentiate the function using the power rule.
3. Simplify the derivative and express it without negative exponents.

Rewrite the function using negative exponents:
$y = x^{-3}$

Differentiate the function using the power rule, which states that if $y = x^n$ , then $\frac{dy}{dx} = nx^{n-1}$ :
$\frac{dy}{dx} = -3x^{-3-1}$ $\frac{dy}{dx} = -3x^{-4}$

Simplify the derivative and express it without negative exponents:
$\frac{dy}{dx} = -\frac{3}{x^4}$
The derivative of the function is:
$\boxed{-\frac{3}{x^4}}$