Calculus: Differentiation and Applications

Vocabulary practice flashcards covering exponential, logarithmic, and trigonometric differentiation rules, along with second derivatives and their applications.

Number $e$

The unique base number for which the limit of the exponential expression as it approaches $0$ is equal to $1$, forming the base of the natural exponential function $y = e^x$.

Derivative of $e^x$

The standard differentiation rule stating that the derivative of the exponential function with base $e$ is itself: $\frac{d}{dx}(e^x) = e^x$.

Derivative of $e^{f(x)}$

The composite differentiation rule stating that $\frac{d}{dx}(e^{f(x)}) = f'(x)e^{f(x)}$.

Natural Logarithmic Function ($y = \ln(x)$)

The logarithmic function with base $e$, written as $y = \ln(x) = \log_e(x)$, which possesses an inverse relationship with the exponential function $y = e^x$.

Derivative of $\ln(x)$

The differentiation rule stating that the derivative of the natural logarithm of $x$ is $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$.

Derivative of $\ln(f(x))$

The chain rule derivative for logarithmic functions stating that $\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$.

Derivative of $\sin(x)$

The basic trigonometric differentiation rule stating that $\frac{d}{dx}(\sin(x)) = \cos(x)$.

Derivative of $\sin(f(x))$

The chain rule derivative for composite sine functions stating that $\frac{d}{dx}(\sin(f(x))) = f'(x)\cos(f(x))$.

Derivative of $\cos(x)$

The basic trigonometric differentiation rule stating that $\frac{d}{dx}(\cos(x)) = -\sin(x)$.

Derivative of $\cos(f(x))$

The chain rule derivative for composite cosine functions stating that $\frac{d}{dx}(\cos(f(x))) = -f'(x)\sin(f(x))$.

Chain Rule

A differentiation method used to determine the derivative of composite functions involving exponential, logarithmic, and trigonometric functions.

Product Rule

A differentiation method used to determine the derivative of a product of functions involving exponential, logarithmic, and trigonometric functions.

Quotient Rule

A differentiation method used to determine the derivative of a quotient of functions involving exponential, logarithmic, and trigonometric functions.

Second Derivative

The derivative of the first derivative function, representing the rate of change of the first derivative.

Acceleration

The physical quantity defined and calculated as the second derivative of displacement position with respect to time.

Concavity and Points of Inflection

Geometric curvature properties and transition points of a function's graph that are directly analyzed through the behavior and sign of the second derivative.

Second Derivative Test

A calculus technique using the value of the second derivative at stationary points to identify and confirm local maxima and local minima.

Optimisation Problems

Applied problems across diverse fields modeled and solved by determining maximum or minimum values using first and second derivatives.