10 Questions
Which of the following is the derivative of $x^n$ with respect to $x$, where $n$ is a constant?
$nx^{n-1}$
What is the derivative of $\sin(x)$ with respect to $x$?
$\cos(x)$
What is the derivative of $\tan(x)$ with respect to $x$?
$\sec^2(x)$
What is the derivative of $\ln(u)$ with respect to $x$, where $u$ is a function of $x$?
$\frac{1}{u} \cdot \frac{du}{dx}$
What is the derivative of $f(g(x))$ with respect to $x$, where $f$ and $g$ are functions?
$f'(g(x)) \cdot g'(x)$
Which of the following is the derivative of $\cot(x)$ with respect to $x$?
$\sec^2(x)$
Which of the following is the derivative of $\sec(x)$ with respect to $x$?
$\sec(x)\tan(x)$
Which of the following is the derivative of $\ln(\ln(x))$ with respect to $x$?
$\frac{1},{x\ln^2(x)}$
Which of the following is the derivative of $\arcsin(\frac{1},{x})$ with respect to $x$?
$\frac{1},{x\sqrt{1-\frac{1},{x^2}}}$
Which of the following is the derivative of $\arccos(x^2)$ with respect to $x$?
$\frac{-2x},{\sqrt{1-x^4}}$
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 $\frac{-1}{\sqrt{1-u^2}}\cdot\frac{du}{dx}$.
Test your knowledge of basic derivatives with this quiz! Explore the derivatives of various functions, including sine, cosine, tangent, cotangent, secant, cosecant, natural logarithm, and more. Improve your understanding of derivatives by practicing with this quiz.
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