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Differentiation of Special Functions

Fill in the blank:
If $y=14t+e^t$ , then $\frac{dy}{dt}=$ __________.

$\frac{14}{t}$

$-\frac{7}{1+e^{t^2}}$

$14+e^t$

Fill in the blank:
$\frac{d}{dv}\left[\cos\cos\frac{v}{3}\right]=$ __________.

$-\sin v$

$-\frac{1}{3}\frac{v}{3}$

$\frac{v}{3}$

Fill in the blank:
If $y=\ln\ln7x$ . Then $\frac{dy}{dx}=$ __________.

$\frac{1}{x}$

$-kx$

$\frac{k}{x}$

Find the derivative of $e^{f\left(x\right)}$ with respect to $x$ .

$\frac{1}{x}$

$e^{f\left(x\right)}$

$f’\left(x\right)e^{f\left(x\right)}$

Fill in the blank:
If $y=\ln\ln\left(f\left(x\right)\right)$ , then $\frac{dy}{dx}=$ __________.

$\frac{f’\left(x\right)}{f\left(x\right)}$

$\frac{\sin t}{1-\cos t}$

$-\frac{f’\left(x\right)}{f\left(x\right)}$

Find the derivative of $\ln\ln\left(3x+11\right)$ with respect to $x$ .

$\frac{3}{\left(3x+11\right)}$

$\frac{x}{\left(2x+1\right)\left(1+x^2\right)}$

$\frac{2x}{\left(2x+1\right)\left(1+x^2\right)}$

Fill in the blank:
If $y=e^{2t}$ , then $\frac{dy}{dt}=$ __________.

$\frac{1}{t}$

$e^{2t}$

$2e^{2t}$

Fill in the blank:
$\frac{d}{dt}\left[\ln\ln11t\right]=$ __________.

$\frac{\sin t}{1-\cos t}$

$-\frac{\sin t}{1-\cos t}$

$\frac{1}{t}$

Differentiate $\sin\left(6x\right)$ with respect to $x$ .

$6\cos\cos6x$

$\cos\cos6x$

$6\csc\csc6x$

Differentiate $\sin\left(5x^2\right)$ with respect to $x$ .

$10x\cos\left(5x^2\right)$

$x\cos\left(5x^2\right)$

$\cos\left(5x^2\right)$

Questions

Time

Score

Differentiation of Special Functions

Fill in the blank:If ﻿y=14t+ety=14t+e^ty=14t+et﻿, then ﻿dydt=\frac{dy}{dt}=dtdy​=﻿ __________.

﻿14t\frac{14}{t}t14​﻿

﻿−71+et2-\frac{7}{1+e^{t^2}}−1+et27​﻿

﻿14+et14+e^t14+et﻿

Fill in the blank:﻿ddv[cos⁡cos⁡v3]=\frac{d}{dv}\left[\cos\cos\frac{v}{3}\right]=dvd​[coscos3v​]=﻿ __________.

﻿−sin⁡v-\sin v−sinv﻿

﻿−13v3-\frac{1}{3}\frac{v}{3}−31​3v​﻿

﻿v3\frac{v}{3}3v​﻿

Fill in the blank:If ﻿y=ln⁡ln⁡7xy=\ln\ln7xy=lnln7x﻿. Then ﻿dydx=\frac{dy}{dx}=dxdy​=﻿ __________.

﻿1x\frac{1}{x}x1​﻿

﻿−kx-kx−kx﻿

﻿kx\frac{k}{x}xk​﻿

Find the derivative of ﻿ef(x)e^{f\left(x\right)}ef(x)﻿ with respect to ﻿xxx﻿.

﻿1x\frac{1}{x}x1​﻿

﻿ef(x)e^{f\left(x\right)}ef(x)﻿

﻿f’(x)ef(x)f’\left(x\right)e^{f\left(x\right)}f’(x)ef(x)﻿

Fill in the blank:If ﻿y=ln⁡ln⁡(f(x))y=\ln\ln\left(f\left(x\right)\right)y=lnln(f(x))﻿, then ﻿dydx=\frac{dy}{dx}=dxdy​=﻿ __________.

﻿f’(x)f(x)\frac{f’\left(x\right)}{f\left(x\right)}f(x)f’(x)​﻿

﻿sin⁡t1−cos⁡t\frac{\sin t}{1-\cos t}1−costsint​﻿

﻿−f’(x)f(x)-\frac{f’\left(x\right)}{f\left(x\right)}−f(x)f’(x)​﻿

Find the derivative of ﻿ln⁡ln⁡(3x+11)\ln\ln\left(3x+11\right)lnln(3x+11)﻿ with respect to ﻿xxx﻿.

﻿3(3x+11)\frac{3}{\left(3x+11\right)}(3x+11)3​﻿

﻿x(2x+1)(1+x2)\frac{x}{\left(2x+1\right)\left(1+x^2\right)}(2x+1)(1+x2)x​﻿

﻿2x(2x+1)(1+x2)\frac{2x}{\left(2x+1\right)\left(1+x^2\right)}(2x+1)(1+x2)2x​﻿

Fill in the blank:If ﻿y=e2ty=e^{2t}y=e2t﻿, then ﻿dydt=\frac{dy}{dt}=dtdy​=﻿ __________.

﻿1t\frac{1}{t}t1​﻿

﻿e2te^{2t}e2t﻿

﻿2e2t2e^{2t}2e2t﻿

Fill in the blank:﻿ddt[ln⁡ln⁡11t]=\frac{d}{dt}\left[\ln\ln11t\right]=dtd​[lnln11t]=﻿ __________.

﻿sin⁡t1−cos⁡t\frac{\sin t}{1-\cos t}1−costsint​﻿

﻿−sin⁡t1−cos⁡t-\frac{\sin t}{1-\cos t}−1−costsint​﻿

﻿1t\frac{1}{t}t1​﻿

Differentiate ﻿sin⁡(6x)\sin\left(6x\right)sin(6x)﻿ with respect to ﻿xxx﻿.

﻿6cos⁡cos⁡6x6\cos\cos6x6coscos6x﻿

﻿cos⁡cos⁡6x\cos\cos6xcoscos6x﻿

﻿6csc⁡csc⁡6x6\csc\csc6x6csccsc6x﻿

Differentiate ﻿sin⁡(5x2)\sin\left(5x^2\right)sin(5x2)﻿ with respect to ﻿xxx﻿.

﻿10xcos⁡(5x2)10x\cos\left(5x^2\right)10xcos(5x2)﻿

﻿xcos⁡(5x2)x\cos\left(5x^2\right)xcos(5x2)﻿

﻿cos⁡(5x2)\cos\left(5x^2\right)cos(5x2)﻿

Incorrect

Great!

Fill in the blank:
If $y=14t+e^t$ , then $\frac{dy}{dt}=$ __________.

$\frac{14}{t}$

$-\frac{7}{1+e^{t^2}}$

$14+e^t$

Fill in the blank:
$\frac{d}{dv}\left[\cos\cos\frac{v}{3}\right]=$ __________.

$-\sin v$

$-\frac{1}{3}\frac{v}{3}$

$\frac{v}{3}$

Fill in the blank:
If $y=\ln\ln7x$ . Then $\frac{dy}{dx}=$ __________.

$\frac{1}{x}$

$-kx$

$\frac{k}{x}$

Find the derivative of $e^{f\left(x\right)}$ with respect to $x$ .

$\frac{1}{x}$

$e^{f\left(x\right)}$

$f’\left(x\right)e^{f\left(x\right)}$

Fill in the blank:
If $y=\ln\ln\left(f\left(x\right)\right)$ , then $\frac{dy}{dx}=$ __________.

$\frac{f’\left(x\right)}{f\left(x\right)}$

$\frac{\sin t}{1-\cos t}$

$-\frac{f’\left(x\right)}{f\left(x\right)}$

Find the derivative of $\ln\ln\left(3x+11\right)$ with respect to $x$ .

$\frac{3}{\left(3x+11\right)}$

$\frac{x}{\left(2x+1\right)\left(1+x^2\right)}$

$\frac{2x}{\left(2x+1\right)\left(1+x^2\right)}$

Fill in the blank:
If $y=e^{2t}$ , then $\frac{dy}{dt}=$ __________.

$\frac{1}{t}$

$e^{2t}$

$2e^{2t}$

Fill in the blank:
$\frac{d}{dt}\left[\ln\ln11t\right]=$ __________.

$\frac{\sin t}{1-\cos t}$

$-\frac{\sin t}{1-\cos t}$

$\frac{1}{t}$

Differentiate $\sin\left(6x\right)$ with respect to $x$ .

$6\cos\cos6x$

$\cos\cos6x$

$6\csc\csc6x$

Differentiate $\sin\left(5x^2\right)$ with respect to $x$ .

$10x\cos\left(5x^2\right)$

$x\cos\left(5x^2\right)$

$\cos\left(5x^2\right)$