Quiz about Differential calculus

Question 1

Given that $$u$$ is a function of $$x$$ , the derivative of $$ an\ u$$ with respect to $$x$$ is

Accepted Answer

$$\sec^2u\frac{du}{dx}$$

Answer

$$\cos^2x\frac{dx}{du}$$

Answer

$$\sec^2x\frac{dx}{du}$$

Answer

$$\cos^2u\frac{du}{dx}$$

Question 2

Find $$\frac{ ext{d}y}{ ext{d}x}$$ , where $$y= an^{-1\ }\frac{x}{a}$$

Accepted Answer

$$\frac{a}{a^2+x^2}$$

Answer

$$\frac{a}{x^2+1}$$

Answer

$$\frac{a}{a^2-x^2}$$

Answer

$$\frac{a}{x^2-a^2}$$

Question 3

Find $$\frac{dy}{dt}$$ where $$y=a\left(1+\cos t\right)$$

Accepted Answer

$$-a\sin t$$

Answer

$$1- an t$$

Answer

$$-a an t$$

Answer

$$1+\sin t$$

Question 4

The derivative of $$m\sin^{-1}x$$ with respect to $$x$$ is

Accepted Answer

$$\frac{m}{\sqrt[]{1-x^2}}$$

Answer

$$\frac{m}{\sqrt[]{1+x^2}}$$

Answer

$$\frac{m}{\sqrt[]{x^2-1}}$$

Answer

$$\frac{m}{\sqrt[]{x^2+1}}$$

Question 5

If $$f\left(x\right)=e^{ax}$$ , then $$f$$ is

Accepted Answer

$$a^2e^{ax}$$

Answer

$$e^2a^{2x}$$

Answer

Answer

$$a^xe^{ax}$$

Question 6

The $$n^{th}$$ derivative of $$\frac{1}{\left(x-1\right)}$$ is

Accepted Answer

$$\frac{n!\left(-1\right)^n}{\left(x-1\right)^{n+1}}$$

Answer

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

Answer

$$\frac{n!\left(1\right)^n}{\left(x-1\right)^{n+1}}$$

Answer

$$\frac{n!\left(-1\right)^n}{\left(x+1\right)^{n+1}}$$

Question 7

If $$u=\frac{xy}{x+y},\ then\ \frac{\partial u}{\partial x}=$$

Accepted Answer

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

Answer

$$\frac{x^2}{\left(x-y\right)^2}$$

Answer

$$\frac{y^2}{\left(x-y\right)^2}$$

Answer

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

Question 8

If $$u=\ln\left( an x+ an y+ an z ight),\ $$ then $$\frac{\partial u}{\partial z}$$ is

Accepted Answer

$$\frac{1}{ an x+ an y+ an z}.\sec^2z$$

Answer

$$\frac{1}{ an x+ an y+ an z}.\cos^2x$$

Answer

$$\frac{1}{ an x+ an y+ an z}. an^2z$$

Answer

$$\frac{1}{ an x+ an y+ an z}.\sec^2y$$

Question 9

Find $$\frac{\partial u}{\partial y}$$ when $$u=\sin^{-1}\left(\frac{y}{x}\right)$$

Accepted Answer

$$\frac{1}{x\ \sqrt[]{1-\frac{y^2}{x^2}}}\ \ \ \ $$

Answer

$$\frac{1}{x\sqrt[]{y-\frac{y^2}{x^2}}}\ \ \ \ $$

Answer

$$\frac{1}{x\ \sqrt[]{1+\frac{x^2}{y^2}}}\ \ \ \ $$

Answer

$$\frac{1}{y\ \sqrt[]{1+\frac{y^2}{x^2}}}\ \ \ \ $$

Question 10

Find $$\frac{\partial u}{\partial z}$$ if $$u=\sin\left(ax+by+cz\right)$$

Accepted Answer

$$c\cos\left(ay+bz+cx\right)$$

Answer

$$a\cos\left(ay+bx+cz\right)$$

Answer

$$b\cos\left(ax+by+cz\right)$$

Answer

$$\cos\left(ax+by+cz\right)$$

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