AP Calculus AB 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 general approach for finding the derivatives of sine, cosine, exponential, and logarithmic functions?
Specific differentiation rules can be used to find the derivatives for these familiar functions without having to use the limit definition of a derivative each time.
Card 1 of 10
All Flashcards (10)
What is the general approach for finding the derivatives of sine, cosine, exponential, and logarithmic functions?
Specific differentiation rules can be used to find the derivatives for these familiar functions without having to use the limit definition of a derivative each time.
Evaluate the limit: $\lim_{h \to 0} \\frac{\\ln(3+h) - \\ln(3)}{h}$
This limit is the definition of the derivative for $f(x) = \\ln x$ at the point $x=3$. Since $f'(x) = \\frac{1}{x}$, the limit evaluates to $f'(3) = \\frac{1}{3}$.
Evaluate the limit: $\lim_{h \to 0} \\frac{e^{1+h} - e}{h}$
This expression represents the derivative of $f(x) = e^x$ at $x=1$. Since $f'(x) = e^x$, the value of the limit is $f'(1) = e^1 = e$.
What is the derivative of the natural exponential function, $f(x) = e^x$?
The derivative of $e^x$ is itself, $e^x$. This is a unique property of the natural exponential function.
What strategy can be used to determine a limit if the expression matches the definition of a derivative?
Recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining the limit by simply evaluating the derivative at the given point.
What is the derivative of $f(x) = \\cos x$?
The derivative of $\\cos x$ is $-\\sin x$. This is a specific rule used for the cosine function.
What is the derivative of the natural logarithmic function, $f(x) = \\ln x$?
The derivative of $\\ln x$ is $\\frac{1}{x}$. This rule applies for $x > 0$.
How can you evaluate the limit $\lim_{h \to 0} \\frac{\\sin(\\frac{\\pi}{2}+h) - \\sin(\\frac{\\pi}{2})}{h}$ by recognizing it as a derivative?
This limit is the definition of the derivative of $f(x) = \\sin x$ at $x = \\frac{\\pi}{2}$. Since the derivative of $\\sin x$ is $\\cos x$, the limit equals $\\cos(\\frac{\\pi}{2})$, which is 0.
What is the derivative of $f(x) = \\sin x$?
The derivative of $\\sin x$ is $\\cos x$. This is a specific rule used to find the derivative of the sine function.
How can a limit expression be interpreted in the context of derivatives?
A limit in the form $\lim_{h \to 0} \\frac{f(a+h) - f(a)}{h}$ can be interpreted as the definition of the derivative of the function $f(x)$ at the point $x=a$.