Solution - Derivative

n x^{n - 1} + \frac{- 1 \times \frac{d}{d x} [n]}{{(n + 1)}^{2}}

n x^{n - 1}+\frac{-1\times \frac{d}{dx}[n]}{\left(n + 1\right)^{2}}

See steps

Step-by-step explanation

1. Solve derivative

Applying the sum rule of derivatives.

$\frac{d}{d x} [x^{n} + \frac{1}{n + 1}] = \frac{d}{d x} [x^{n}] + \frac{d}{d x} [\frac{1}{n + 1}]$

Computing the derivative of x raised to the power of n.

$\frac{d}{d x} [x^{n}] + \frac{d}{d x} [\frac{1}{n + 1}] = n x^{n - 1} + \frac{d}{d x} [\frac{1}{n + 1}]$

Computing the derivative of a fraction.

$n x^{n - 1} + \frac{d}{d x} [\frac{1}{n + 1}] = n x^{n - 1} + \frac{\frac{d}{d x} [1] \times (n + 1) - 1 \times \frac{d}{d x} [n + 1]}{{(n + 1)}^{2}}$

The derivative of a constant value is always zero.

$n x^{n - 1} + \frac{\frac{d}{d x} [1] \times (n + 1) - 1 \times \frac{d}{d x} [n + 1]}{{(n + 1)}^{2}} = n x^{n - 1} + \frac{0 (n + 1) - 1 \times \frac{d}{d x} [n + 1]}{{(n + 1)}^{2}}$

Applying the sum rule of derivatives.

$n x^{n - 1} + \frac{0 \times (n + 1) - 1 \times \frac{d}{d x} [n + 1]}{{(n + 1)}^{2}} = n x^{n - 1} + \frac{0 \times (n + 1) - 1 (\frac{d}{d x} [n] + \frac{d}{d x} [1])}{{(n + 1)}^{2}}$

Multiplying a number by zero always results in zero.

$n x^{n - 1} + \frac{0 \times (n + 1) - 1 (\frac{d}{d x} [n] + \frac{d}{d x} [1])}{{(n + 1)}^{2}} = n x^{n - 1} + \frac{0 - 1 (\frac{d}{d x} [n] + \frac{d}{d x} [1])}{{(n + 1)}^{2}}$

The derivative of a constant value is always zero.

$n x^{n - 1} + \frac{0 - 1 (\frac{d}{d x} [n] + \frac{d}{d x} [1])}{{(n + 1)}^{2}} = n x^{n - 1} + \frac{0 - 1 (\frac{d}{d x} [n] + 0)}{{(n + 1)}^{2}}$

Adding zero to a number, which does not change its value.

$n x^{n - 1} + \frac{0 - 1 (\frac{d}{d x} [n] + 0)}{{(n + 1)}^{2}} = n x^{n - 1} + \frac{- 1 (\frac{d}{d x} [n] + 0)}{{(n + 1)}^{2}}$

Adding zero to a number, which does not change its value.

$n x^{n - 1} + \frac{- 1 (\frac{d}{d x} [n] + 0)}{{(n + 1)}^{2}} = n x^{n - 1} + \frac{- 1 \times \frac{d}{d x} [n]}{{(n + 1)}^{2}}$

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Why learn this

Ever wondered how to predict the future? Derivatives are your crystal ball!

Picture this: You're a surfer trying to catch the biggest wave. How do you know when it's coming? Derivatives can tell you when it's at its highest point!

Rocket Science: Planning to launch a rocket to Mars? Derivatives tell us the optimal fuel burn rate to minimize fuel consumption and maximize distance!

Stock Market: Trading in the stock market? Derivatives can indicate the rate at which stock prices are changing, helping predict the best time to buy or sell.

Animation: Love animated movies? Artists use derivatives to smoothly change the motion and expressions of characters, making them feel more lifelike.

Engineering: Designing a bridge or a skyscraper? Derivatives help determine the rates of stress and strain changes in materials, ensuring the safety of your structures.

In short, derivatives are like a secret code to understanding change and making predictions in real life. So let's crack this code together and become masters of our futures!

Terms and topics

Derivative

Solution - Derivative

Step-by-step explanation

1. Solve derivative

Why learn this

Terms and topics

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