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Mastering Basic Derivatives

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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?

$nx^{n-1}$ (correct)

$n^2x^n$

$n^2x^{n-1}$

$nx^n$

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 $x$?

$\sec^2(x)$ (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 $x$?

$\frac{1}{u} \cdot \frac{du}{dx}$ Signup and view all the answers

What is the derivative of $f(g(x))$ with respect to $x$, where $f$ and $g$ are functions?

$f'(g(x)) \cdot g'(x)$ Signup and view all the answers

Which of the following is the derivative of $\cot(x)$ with respect to $x$?

$\sec^2(x)$ Signup and view all the answers

Which of the following is the derivative of $\sec(x)$ with respect to $x$?

$\sec(x)\tan(x)$ Signup and view all the answers

Which of the following is the derivative of $\ln(\ln(x))$ with respect to $x$?

$\frac{1},{x\ln^2(x)}$ Signup and view all the answers

Which of the following is the derivative of $\arcsin(\frac{1},{x})$ with respect to $x$?

$\frac{1},{x\sqrt{1-\frac{1},{x^2}}}$ Signup and view all the answers

Which of the following is the derivative of $\arccos(x^2)$ with respect to $x$?

$\frac{-2x},{\sqrt{1-x^4}}$ Signup and view all the answers

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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Description

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.