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Solution - Derivative

\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + 0 + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + 20 e n s x

\frac{d}{dx}[s]\times e\times n\times (10 x^{2})+0+s\times e\times \frac{d}{dx}[n]\times (10 x^{2})+20 e n s x

See steps

Step-by-step explanation

1. Solve derivative

19 additional steps

Expanding the derivative for multiplication.

$\frac{d}{d x} [s \times e \times n \times (10 x^{2})] = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times \frac{d}{d x} [10 x^{2}]$

Expanding the derivative for multiplication.

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [s \times e \times n \times (10 x^{2})] = \frac{d}{d x} [s \times (e \times n \times (10 x^{2}))]$

Applying the product rule of derivatives.

$\frac{d}{d x} [s \times (e \times n \times (10 x^{2}))] = \frac{d}{d x} [s] \times (e \times n \times (10 x^{2})) + s \times \frac{d}{d x} [e \times n \times (10 x^{2})]$

Expanding the derivative for multiplication.

$\frac{d}{d x} [s] \times (e \times n \times (10 x^{2})) + s \times \frac{d}{d x} [e \times n \times (10 x^{2})] = \frac{d}{d x} [s] \times (e \times n \times (10 x^{2})) + s (\frac{d}{d x} [e] \times n \times (10 x^{2}) + e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times n \times \frac{d}{d x} [10 x^{2}])$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [e \times n \times (10 x^{2})] = \frac{d}{d x} [e \times (n \times (10 x^{2}))]$

Applying the product rule of derivatives.

$\frac{d}{d x} [e \times (n \times (10 x^{2}))] = \frac{d}{d x} [e] \times (n \times (10 x^{2})) + e \times \frac{d}{d x} [n \times (10 x^{2})]$

Expanding the derivative for multiplication.

Applying the product rule of derivatives.

$\frac{d}{d x} [n \times (10 x^{2})] = \frac{d}{d x} [n] \times (10 x^{2}) + n \times \frac{d}{d x} [10 x^{2}]$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [e] \times (n \times (10 x^{2})) + e \times (\frac{d}{d x} [n] \times (10 x^{2}) + n \times \frac{d}{d x} [10 x^{2}]) = \frac{d}{d x} [e] \times n \times (10 x^{2}) + e \times (\frac{d}{d x} [n] \times (10 x^{2}) + n \times \frac{d}{d x} [10 x^{2}])$

Multiplying a number by a sum or difference of two numbers can be done by multiplying each number individually and then adding or subtracting the results.

$\frac{d}{d x} [e] \times n \times (10 x^{2}) + e \times (\frac{d}{d x} [n] \times (10 x^{2}) + n \times \frac{d}{d x} [10 x^{2}]) = \frac{d}{d x} [e] \times n \times (10 x^{2}) + (e \times (\frac{d}{d x} [n] \times (10 x^{2})) + e \times (n \times \frac{d}{d x} [10 x^{2}]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [e] \times n \times (10 x^{2}) + (e \times (\frac{d}{d x} [n] \times (10 x^{2})) + e \times (n \times \frac{d}{d x} [10 x^{2}])) = \frac{d}{d x} [e] \times n \times (10 x^{2}) + (e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times (n \times \frac{d}{d x} [10 x^{2}]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [e] \times n \times (10 x^{2}) + (e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times (n \times \frac{d}{d x} [10 x^{2}])) = \frac{d}{d x} [e] \times n \times (10 x^{2}) + (e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times n \times \frac{d}{d x} [10 x^{2}])$

Addition can be grouped differently, but the result remains the same.

$\frac{d}{d x} [e] \times n \times (10 x^{2}) + (e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times n \times \frac{d}{d x} [10 x^{2}]) = \frac{d}{d x} [e] \times n \times (10 x^{2}) + e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times n \times \frac{d}{d x} [10 x^{2}]$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [s] \times (e \times n \times (10 x^{2})) + s (\frac{d}{d x} [e] \times n \times (10 x^{2}) + e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times n \times \frac{d}{d x} [10 x^{2}]) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s (\frac{d}{d x} [e] \times n \times (10 x^{2}) + e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times n \times \frac{d}{d x} [10 x^{2}])$

Multiplying a number by a sum or difference of two numbers can be done by multiplying each number individually and then adding or subtracting the results.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s (\frac{d}{d x} [e] \times n \times (10 x^{2}) + e \times \frac{d}{d x} [n] \times (10 x^{2}) + e \times n \times \frac{d}{d x} [10 x^{2}]) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + (s \times (\frac{d}{d x} [e] \times n \times (10 x^{2})) + s \times (e \times \frac{d}{d x} [n] \times (10 x^{2})) + s \times (e \times n \times \frac{d}{d x} [10 x^{2}]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + (s \times (\frac{d}{d x} [e] \times n \times (10 x^{2})) + s \times (e \times \frac{d}{d x} [n] \times (10 x^{2})) + s \times (e \times n \times \frac{d}{d x} [10 x^{2}])) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + (s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times (e \times \frac{d}{d x} [n] \times (10 x^{2})) + s \times (e \times n \times \frac{d}{d x} [10 x^{2}]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + (s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times (e \times \frac{d}{d x} [n] \times (10 x^{2})) + s \times (e \times n \times \frac{d}{d x} [10 x^{2}])) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + (s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times (e \times n \times \frac{d}{d x} [10 x^{2}]))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + (s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times (e \times n \times \frac{d}{d x} [10 x^{2}])) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + (s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times \frac{d}{d x} [10 x^{2}])$

Addition can be grouped differently, but the result remains the same.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + (s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times \frac{d}{d x} [10 x^{2}]) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times \frac{d}{d x} [10 x^{2}]$

The derivative of a constant value is always zero.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times \frac{d}{d x} [e] \times n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times \frac{d}{d x} [10 x^{2}] = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times \frac{d}{d x} [10 x^{2}]$

Applying the product rule of derivatives.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times \frac{d}{d x} [10 x^{2}] = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n (\frac{d}{d x} [10] \times x^{2} + 10 \times \frac{d}{d x} [x^{2}])$

The derivative of a constant value is always zero.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n (\frac{d}{d x} [10] \times x^{2} + 10 \times \frac{d}{d x} [x^{2}]) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n (0 x^{2} + 10 \times \frac{d}{d x} [x^{2}])$

Multiplying a number by zero always results in zero.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n (0 x^{2} + 10 \times \frac{d}{d x} [x^{2}]) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n (0 + 10 \times \frac{d}{d x} [x^{2}])$

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

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n (0 + 10 \times \frac{d}{d x} [x^{2}]) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (10 \times \frac{d}{d x} [x^{2}])$

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

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (10 \times \frac{d}{d x} [x^{2}]) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (10 \times (2 x^{2 - 1}))$

Subtracting one from a number.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (10 \times (2 x^{2 - 1})) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (10 \times (2 x^{1}))$

Multiplication can be grouped differently, but the result remains the same.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (10 \times (2 x^{1})) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times ((10 \times 2) \times x^{1})$

Multiplying two integers together.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times ((10 \times 2) \times x^{1}) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (20 x^{1})$

Simplifying the arithmetic expressions.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (20 x^{1}) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (20 x)$

Simplifying the arithmetic expressions.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + s \times 0 n \times (10 x^{2}) + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (20 x) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + 0 + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (20 x)$

Simplifying the arithmetic expressions.

$\frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + 0 + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + s \times e \times n \times (20 x) = \frac{d}{d x} [s] \times e \times n \times (10 x^{2}) + 0 + s \times e \times \frac{d}{d x} [n] \times (10 x^{2}) + 20 e n s x$

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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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