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Questions and Answers
Which of the following is the derivative of $x^n$ with respect to $x$, where $n$ is a constant?
Which of the following is the derivative of $x^n$ with respect to $x$, where $n$ is a constant?
- $nx^{n-1}$ (correct)
- $n^2x^n$
- $n^2x^{n-1}$
- $nx^n$
What is the derivative of $\sin(x)$ with respect to $x$?
What is the derivative of $\sin(x)$ with respect to $x$?
- $-\sin(x)$
- $\sin(x)$
- $-\cos(x)$
- $\cos(x)$ (correct)
What is the derivative of $\tan(x)$ with respect to $x$?
What is the derivative of $\tan(x)$ with respect to $x$?
- $\sec^2(x)$ (correct)
- $-\sec^2(x)$
- $-\csc^2(x)$
- $\csc^2(x)$
What is the derivative of $\ln(u)$ with respect to $x$, where $u$ is a function of $x$?
What is the derivative of $\ln(u)$ with respect to $x$, where $u$ is a function of $x$?
What is the derivative of $f(g(x))$ with respect to $x$, where $f$ and $g$ are functions?
What is the derivative of $f(g(x))$ with respect to $x$, where $f$ and $g$ are functions?
Which of the following is the derivative of $\cot(x)$ with respect to $x$?
Which of the following is the derivative of $\cot(x)$ with respect to $x$?
Which of the following is the derivative of $\sec(x)$ with respect to $x$?
Which of the following is the derivative of $\sec(x)$ with respect to $x$?
Which of the following is the derivative of $\ln(\ln(x))$ with respect to $x$?
Which of the following is the derivative of $\ln(\ln(x))$ with respect to $x$?
Which of the following is the derivative of $\arcsin(\frac{1},{x})$ with respect to $x$?
Which of the following is the derivative of $\arcsin(\frac{1},{x})$ with respect to $x$?
Which of the following is the derivative of $\arccos(x^2)$ with respect to $x$?
Which of the following is the derivative of $\arccos(x^2)$ with respect to $x$?
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Study Notes
Derivatives of Various Functions
- The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$, where $n$ is a constant.
Trigonometric Functions
- The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$.
- The derivative of $\tan(x)$ with respect to $x$ is $\sec^2(x)$.
- The derivative of $\cot(x)$ with respect to $x$ is $-\csc^2(x)$.
- The derivative of $\sec(x)$ with respect to $x$ is $\sec(x)\tan(x)$.
Logarithmic Functions
- The derivative of $\ln(u)$ with respect to $x$, where $u$ is a function of $x$, is $\frac{1}{u}\cdot\frac{du}{dx}$.
- The derivative of $\ln(\ln(x))$ with respect to $x$ is $\frac{1}{x\ln(x)}$.
Composite Functions
- The derivative of $f(g(x))$ with respect to $x$, where $f$ and $g$ are functions, is $f'(g(x))\cdot g'(x)$.
Inverse Trigonometric Functions
- The derivative of $\arcsin(u)$ with respect to $x$, where $u$ is a function of $x$, is $\frac{1}{\sqrt{1-u^2}}\cdot\frac{du}{dx}$.
- The derivative of $\arccos(u)$ with respect to $x$, where $u$ is a function of $x$, is $\frac{-1}{\sqrt{1-u^2}}\cdot\frac{du}{dx}$.
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