Derivatives of Exponential and Logarithmic Functions
It is the derivative of e^x.
What is e^x?
It is the derivative of \sin x.
What is \cos x?
It is the derivative of f(x) g(x).
What is f(x) g'(x) + f'(x) g(x)?
It is the derivative of f(g(x)).
What is f'(g(x)) g'(x)?
It is the slope of the tangent line to x + y = 1 at the point (\frac 1 2, \frac 1 2).
What is \frac{dy}{dx} = -1?
It is the derivative of y if y = f(x) is a function of x.
What is y' or f'(x) or \frac{dy}{dx}?
It is the derivative of \ln x.
What is \frac 1 x?
It is the derivative of \tan x.
What is \sec^2 x?
It is the derivative of \frac{f(x)}{g(x)}.
What is \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}?
It is the derivative of f(g(h(x))).
What is f'(g(h(x))) g'(h(x)) h'(x)?
It is the slope of the tangent line to x \sin y = -1 at the point (1, \frac{3 \pi} 2).
What is \frac{dy}{dx} is undefined?
It is the acceleration of an object in motion with position function f(t) = t + \sqrt t.
What is a(t) = -\frac 1 {4 t^{3/2}}?
It is the derivative of b^x + \log_b x.
What is b^x \ln b + \frac 1 {x \ln b}?
It is the derivative of \tan^{-1} x.
What is \frac 1 {x^2 + 1}?
It is the derivative of x^3 e^x.
What is x^3 e^x + 3x^2 e^x?
It is the derivative of (x^3 + x^2 + 1)^{100}.
What is 100(x^3 + x^2 + 1)^{99} (3x^2 + 2x)?
It is the point-slope form of the tangent line to \frac{x + y}{y^2 + 1} = 2 at the point (2, 0).
What is y = -(x - 2)?
It is the instantaneous rate of change of the area a of a rectangle whose length l is twice its width w.
What is \frac{da}{dt} = 4w \frac{dw}{dt}?
It is the derivative of x \ln x.
What is \ln x + 1?
It is the derivative of x^2 \sin^{-1} x.
What is 2x \sin^{-1} x + \frac{x^2}{\sqrt{1 - x^2}}?
It is the derivative of \frac{x^3}{\ln x}.
What is \frac{3x^2 \ln x - x^2}{\ln^2 x}?
It is the derivative of \sin^3 x.
What is 3 \sin^2 x \cos x?
It is the point-slope form of the tangent line to e^{xy^2} = e at the point (1, 1).
What is y - 1 = -\frac 1 2 (x - 1)?
It is the instantaneous rate of change of the radius r of a circle whose area a increases at a rate of 1 \frac{text{cm}^2}{\text{s}} at the moment the circumference is 2 cm.
What is \frac 1 2 \frac{\text{cm}}{\text{s}}?
It is the derivative of \frac{x + e^x}{\ln x + 1}.
What is \frac{(1 + e^x) \ln x - \frac 1 x (x + e^x)}{(\ln x + 1)^2}?
It is the derivative of \sec^{-1}(\ln x).
What is \frac 1 {\ln x \sqrt{\ln^2 x - 1}}?
It is the derivative of \frac{x^2 \sin x}{\sec x + 1}.
What is \frac{(x^2 \cos x + 2x \sin x)(\sec x + 1) - x^2 \sin x \sec x \tan x}{(\sec x + 1)^2}?
It is the derivative of \tan^2(x^2).
What is 2 \tan(x^2) \sec^2(x^2) (2x)?
It is the point-slope form of the tangent line to y = \sin^x x at x = 1.
What is y - \sin 1 = \sin 1 [\cot 1 + \ln(\sin 1)](x - 1)?
It is the instantaneous rate of change of the water depth h of a right-circular cone of radius r = 6 ft and height h = 12 ft whose volume V decreases at a rate of 2 \frac{text{ft}^3}{\text{s}} at the moment the water depth is 3 ft.
What is \frac{dh}{dt} = -\frac 4 {3 \pi}?