Mathematical Mastery

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What is the derivative of the function $f(x) = x\sqrt{x^2-1}$?

$f'(x) = -x\sqrt{x^2-1}$

What is the derivative of the function $g(x) = \sin^{-1}(x)e^x(x+2)$?

$g'(x) = \frac{d}{dx}[\sin^{-1}(x)e^x(x+2)] = \frac{d}{dx}\sin^{-1}(x) \cdot e^x(x+2) + \sin^{-1}(x) \cdot \frac{d}{dx}[e^x(x+2)]$$g'(x) = \frac{1}{\sqrt{1-x^2}} \cdot e^x(x+2) + \sin^{-1}(x) \cdot \left(e^x \cdot \frac{d}{dx}(x+2) + (x+2) \cdot \frac{d}{dx}e^x\right)$$g'(x) = \frac{1}{\sqrt{1-x^2}} \cdot e^x(x+2) + \sin^{-1}(x) \cdot \left(e^x + (x+2) \cdot e^x\right)$

What is the derivative of the function $h(x) = \frac{d}{dx}(x^2 \cos(\theta))$?

$h'(x) = \frac{d}{dx}(x^2 \cos(\theta)) = \frac{d}{dx}x^2 \cdot \cos(\theta) + x^2 \cdot \frac{d}{dx}\cos(\theta)$$h'(x) = 2x \cos(\theta) + x^2 \cdot (-\sin(\theta) \cdot \frac{d\theta}{dx})$

What is the second derivative of the function $f(x) = x^3 - x^2 + 1$?

$f''(x) = \frac{d^2}{dx^2}(x^3 - x^2 + 1) = \frac{d}{dx}(3x^2 - 2x) = 6x - 2$

What is the derivative of the function $g(x) = \frac{1}{x^2}$?

$g'(x) = \frac{d}{dx}\left(\frac{1}{x^2}\right) = -\frac{2}{x^3}$

Study Notes

Key Concepts in Trigonometric Functions and Derivatives

  • The text discusses various mathematical concepts, including functions, inverse functions, logarithmic functions, implicit functions, and parametric functions.
  • It mentions the derivative of a function with respect to x, denoted as dy/dx, and the derivative of the sine function.
  • The formula for the second derivative of a function is given as -x√(x^2-1).
  • The text also mentions trigonometric functions like secant (secx), cosecant (cosecx), and their derivatives.
  • The derivative of the inverse function is given as (d/dx)(f^(-1)(x)) = (1/f'(f^(-1)(x))).
  • The derivative of the function e^x * (x + 2) is mentioned.
  • The concept of implicit functions and their derivatives is discussed.
  • The text presents a formula involving theta and the derivative of x^2 * square(theta).
  • The derivative of 2x^5 - 1/3 is given as 10x^4 - 3.
  • The concept of inverse functions is mentioned, particularly in relation to trigonometric functions.
  • The text discusses the derivative of the function x^2/(x^2-1).
  • The concept of zero order derivatives and their relation to x^3 and x^2 is mentioned.

Test your knowledge on foreign functions, derivatives, and logarithmic functions with this challenging quiz. Explore topics such as inverse functions, parametric functions, and more. Put your math skills to the test and see how well you can differentiate and derive equations.

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