AP Calculus BC Flashcards: Derivatives of $\cos x$, $\sin x$, $e^x$, and $\ln x$
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is the specific rule for the derivative of $\sin x$?
The derivative of $\sin x$ is $\cos x$. This is a specific rule used to find the derivative of the sine function.
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What is the specific rule for the derivative of $\sin x$?
The derivative of $\sin x$ is $\cos x$. This is a specific rule used to find the derivative of the sine function.
How can recognizing the definition of a derivative be a useful strategy?
It offers a strategy for determining the value of a complex limit by identifying it as the derivative of a known function at a specific point.
What is the specific rule for the derivative of $\cos x$?
The derivative of $\cos x$ is $-\sin x$. This is a specific rule used to find the derivative of the cosine function.
What is the specific rule for the derivative of the natural logarithmic function, $\ln x$?
The derivative of $\ln x$ is $\frac{1}{x}$. This rule applies for $x > 0$.
What is the main idea behind finding derivatives for functions like sine, cosine, exponential, and logarithmic functions?
The main idea is that specific, established rules can be used to find the derivatives for these familiar functions without having to use the limit definition every time.
What is the specific rule for the derivative of the natural exponential function, $e^x$?
The derivative of $e^x$ is $e^x$. The exponential function is its own derivative.
What is the primary skill involved in calculating derivatives of familiar functions?
The primary skill is the application of specific rules for common functions such as sine, cosine, exponential, and logarithmic functions.
How would you interpret the limit $\lim_{h \to 0} \frac{\cos(\pi+h) - \cos(\pi)}{h}$?
This limit is the definition of the derivative of the function $f(x) = \cos x$ at the point $x = \pi$.
To find the value of the limit $\lim_{h \to 0} \frac{e^{2+h} - e^2}{h}$, what function's derivative should you calculate?
You should calculate the derivative of the function $f(x) = e^x$ and evaluate it at $x=2$.
What is the relationship between a limit and a derivative?
A derivative is fundamentally defined as a specific type of limit. This limit represents the instantaneous rate of change of a function.