Solution - Derivative

- \frac{\cos (x)}{\sin (x)}

- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

See steps

Step-by-step explanation

1. Solve derivative

Applying the product rule of derivatives.

$\frac{d}{d x} [- 1 \times \ln (\sin (x))] = \frac{d}{d x} [- 1] \times \ln (\sin (x)) - 1 \times \frac{d}{d x} [\ln (\sin (x))]$

The derivative of a constant value is always zero.

$\frac{d}{d x} [- 1] \times \ln (\sin (x)) - 1 \times \frac{d}{d x} [\ln (\sin (x))] = 0 \times \ln (\sin (x)) - 1 \times \frac{d}{d x} [\ln (\sin (x))]$

Multiplying a number by zero always results in zero.

$0 \times \ln (\sin (x)) - 1 \times \frac{d}{d x} [\ln (\sin (x))] = 0 - 1 \times \frac{d}{d x} [\ln (\sin (x))]$

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

$0 - 1 \times \frac{d}{d x} [\ln (\sin (x))] = - 1 \times \frac{d}{d x} [\ln (\sin (x))]$

2 additional steps

Computing the derivative of a logarithm function using the chain rule.

$- 1 \times \frac{d}{d x} [\ln (\sin (x))] = - 1 \times (\frac{1}{\sin (x)} \times \frac{d}{d x} [\sin (x)])$

Decomposing the function for the chain rule.

$\frac{d}{d x} [\ln (\sin (x))] = \frac{d}{d x} [\ln (x)] \times \frac{d}{d x} [\sin (x)]$

Computing the derivative of a natural logarithm function.

$\frac{d}{d x} [\ln (x)] \times \frac{d}{d x} [\sin (x)] = \frac{1}{x} \times \frac{d}{d x} [\sin (x)]$

Substituting the variable back into the function.

$\frac{1}{x} \times \frac{d}{d x} [\sin (x)] = \frac{1}{\sin (x)} \times \frac{d}{d x} [\sin (x)]$

Computing the derivative of a sine function.

$- 1 \times (\frac{1}{\sin (x)} \times \frac{d}{d x} [\sin (x)]) = - 1 \times (\frac{1}{\sin (x)} \times \cos (x))$

Simplifying the arithmetic expressions.

$- 1 \times (\frac{1}{\sin (x)} \times \cos (x)) = - 1 \times (\frac{\cos (x)}{\sin (x)})$

Simplifying the arithmetic expressions.

$- 1 \times (\frac{\cos (x)}{\sin (x)}) = - \frac{\cos (x)}{\sin (x)}$

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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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