1. Apply the constant multiple rule and derivative of $\sin x$. 2. Apply the power rule to $2x^2$. 3. Combine the results: $f'(x) = 3\cos x + 4x$.

1. Apply the constant multiple rule and derivative of $e^x$. 2. Apply the derivative of $\cos x$. 3. Combine the results: $f'(x) = 5e^x + \sin x$.

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: $f'(x) = \frac{2}{x} + 4\cos x$.

$e^x$ is a unique function where its rate of change is always proportional to its current value.

It indicates that as $x$ increases, $\cos x$ decreases in the first quadrant, hence the negative rate of change.

They are fundamental building blocks used in conjunction with other rules like the product, quotient, and chain rules.

The instantaneous rate of change of a function with respect to its variable.

A trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right triangle.

A trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right triangle.

The exponential function with base $e$, where $e$ is Euler's number (approximately 2.71828).

The natural logarithm function, which is the logarithm to the base $e$.