Differentiation of Exponential Functions Quiz

Derivatives

• Derivative of $a^x$ is $a^x \ln a$
• derivative of $x^4 + 2x^2 + 2$ is $4x\sqrt{y-1}$
• derivative of $(1+x^2)^n$ w.r.t $x^2$ is $n(1+x^2)^{n-1}2x$
• derivative of $xy + y^2 = 2$ is $\frac{-y}{x + 2y}$
• derivative of $\sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x}}}+\ldots \infty$ is $\frac{\sec^2x}{2y-1}$
• derivative of $\tan (ptan^{-1}x)$ is $\frac{(1+x^2)y^2 - p(1+y^2)}{(1+x^2)y}$
• derivative of $\frac{1}{a} \sin^{-1} \frac{a}{x}$ is $\frac{1}{\sqrt{x^2-a^2}}$
• derivative of $x = \theta + \frac{1}{\theta}, y = \theta + 1$ is $\frac{dy}{dx} = \frac{y-1}{x-1}$
• derivative of $\tan^{-1}x$ is $\frac{1}{1+x^2}$
• derivative of $y = x^4 + 2x^2 + 2$ is $4x\sqrt{y-1}$
• derivative of $x = \frac{1-t^2}{1+t^2}, y = \frac{2t}{1+t^2}$ is $y\frac{dy}{dx} + x = 0$
• derivative of $\frac{y}{x} = \tan^{-1} \frac{x}{y}$ is $\frac{dy}{dx} = \frac{y}{x}$

Types of Functions

• Derivative of a cubic function is a quadratic function

Trigonometric Functions

• Derivative of $\tan x$ is $\sec^2 x$
• Derivative of $\sin x$ is $\cos x$
• Derivative of $\cos x$ is $-\sin x$
• Derivative of $\sinh 2x$ is $2\cosh 2x$

Other Functions

• Derivative of $\frac{2}{\sqrt{x}}$ is $-\frac{1}{x\sqrt{x}}$
• Derivative of $\cos x^0$ is $0$
• Derivative of $(x^3+1)^9$ w.r.t $x^2$ is $27x(x^3+1)^8$