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Questions and Answers
What is the derivative of the function $f(x) = x\sqrt{x^2-1}$?
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 $g(x) = \sin^{-1}(x)e^x(x+2)$?
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 $h(x) = \frac{d}{dx}(x^2 \cos(\theta))$?
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 $f(x) = x^3 - x^2 + 1$?
What is the second derivative of the function $f(x) = x^3 - x^2 + 1$?
What is the derivative of the function $g(x) = \frac{1}{x^2}$?
What is the derivative of the function $g(x) = \frac{1}{x^2}$?
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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.
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