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What is a derivative?
The instantaneous rate of change of a function with respect to its variable.
Define $\sin x$.
A trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right triangle.
Define $\cos x$.
A trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right triangle.
Define $e^x$.
The exponential function with base $e$, where $e$ is Euler's number (approximately 2.71828).
Define $\ln x$.
The natural logarithm function, which is the logarithm to the base $e$.
Why is the derivative of $e^x$ equal to itself?
$e^x$ is a unique function where its rate of change is always proportional to its current value.
Explain the significance of the negative sign in the derivative of $\cos x$.
It indicates that as $x$ increases, $\cos x$ decreases in the first quadrant, hence the negative rate of change.
Why is understanding these derivatives crucial for more complex calculus problems?
They are fundamental building blocks used in conjunction with other rules like the product, quotient, and chain rules.
How to find the derivative of $f(x) = 3\sin x + 2x^2$?
1. Apply the constant multiple rule and derivative of $\sin x$. 2. Apply the power rule to $2x^2$. 3. Combine the results: $f'(x) = 3\cos x + 4x$.
Steps to find the derivative of $f(x) = 5e^x - \cos x$?
1. Apply the constant multiple rule and derivative of $e^x$. 2. Apply the derivative of $\cos x$. 3. Combine the results: $f'(x) = 5e^x + \sin x$.
How to find the derivative of $f(x) = 2\ln x + 4\sin x$?
1. Apply the constant multiple rule and derivative of $\ln x$. 2. Apply the constant multiple rule and derivative of $\sin x$. 3. Combine the results: $f'(x) = \frac{2}{x} + 4\cos x$.