Quiz about Product & Quotient Derivative Rules

Question 1

$$\frac{d}{dx}\left[\sin\left(x\right)e^x\right]$$

Accepted Answer

$$\sin\left(x\right)e^x+\cos\left(x\right)e^x$$

Answer

$$\sin\left(x\right)e^{x\ }-\cos\left(x\right)e^x$$

Answer

$$\sin\left(x\right)e^x-x\cos\left(x\right)e^x$$

Answer

None of the above

Question 2

$$\frac{d}{dx}\left[x^7\left(1-\cos\left(x\right)\right)\right]$$

Accepted Answer

None of the above

Answer

$$x^7\sin\left(x\right)-7x^6\cos\left(x\right)$$

Answer

$$-x^7\sin\left(x\right)+7x^6\left(1-\cos\left(x\right)\right)$$

Answer

$$x^7\left(1-\cos\left(x\right)\right)+7x^6\sin\left(x\right)$$

Question 3

$$\frac{d}{dx}\left[\frac{\ln\left(x\right)}{3-e^x}\right]$$

Accepted Answer

$$\frac{\frac{\left(3-e^x\right)}{x}+\ln\left(x\right)e^x}{\left(3-e^x\right)^2}$$

Answer

$$\frac{\frac{\left(3-e^x\right)}{x}-\ln\left(x\right)e^x}{\left(3-e^x\right)^2}$$

Answer

$$\frac{x\left(3-e^x\right)+\ln\left(x\right)e^x}{\left(3-e^x\right)^2}$$

Answer

None of the above

Question 4

$$\frac{d}{dx}\left[\frac{x^3}{\sqrt[]{x}}\cos\left(x\right)\right]$$

Accepted Answer

$$-x^{\frac{5}{2}}\sin\left(x\right)+\frac{5}{2}x^{\frac{3}{2}}\cos\left(x\right)$$

Answer

$$-\frac{x^3}{\sqrt[]{x}}\sin\left(x\right)+\frac{3x^2}{\frac{1}{2}\sqrt[]{x}}\cos\left(x\right)$$

Answer

$$-x^{\frac{7}{2}}\sin\left(x\right)+\frac{7}{2}x^{\frac{5}{2}}\cos\left(x\right)$$

Answer

None of the above

Question 5

$$\frac{d}{dx}\left[\frac{3x^4}{2e^x}\right]$$

Accepted Answer

$$\frac{24x^3e^{x\ }-\ 6x^4e^x}{4e^{2x}}$$

Answer

$$\frac{24x^3e^{x\ }+\ 6x^4e^x}{4e^{2x}}$$

Answer

$$\frac{4x^3e^{x\ }-\ 2x^4e^x}{4e^{2x}}$$

Question 6

$$\frac{d}{dx}\ln\left(x\right)8^x$$

Accepted Answer

None of the above

Answer

$$\ln\left(x\right)8^x+x8^x$$

Answer

$$\frac{8^x\ln\left(8\right)}{x}+\ln\left(x\right)8^x$$

Answer

$$\ln\left(x\right)8^x\ln\left(8\right)+x8^x$$

Question 7

$$\frac{d}{dx}\left[\frac{-4\sin\left(x\right)}{\sqrt[5]{x^2}}\right]$$

Accepted Answer

$$\frac{-4\sqrt[5]{x^2}\cos\left(x\right)+\frac{8\sin\left(x\right)}{5\sqrt[5]{x^3}}}{\sqrt[5]{x^4}}$$

Answer

$$\frac{4\sqrt[5]{x^2}\cos\left(x\right)-\frac{8\sin\left(x\right)}{5\sqrt[5]{x^3}}}{\sqrt[5]{x^4}}$$

Answer

$$\frac{-4\sqrt[5]{x^2}\cos\left(x\right)+\frac{10\sin\left(x\right)}{\sqrt[3]{x^2}}}{\sqrt[5]{x^4}}$$

Answer

None of the above

Question 8

$$\frac{d}{dx}\left[\left(14+3x^2\right)\left(3^x-\cos\left(x\right)\right)\right]$$

Accepted Answer

$$\left(14+3x^2\right)\left(3^x\ln\left(3\right)+\sin\left(x\right)\right)+6x\left(3^x-\cos\left(x\right)\right)$$

Answer

$$\left(6x\right)\left(3^x\ln\left(3\right)-\sin\left(x\right)\right)+\left(14+3x^2\right)\left(3^{x\ }-\cos\left(x\right)\right)$$

Answer

$$\left(14+3x^2\right)\left(3^x\ln\left(3\right)-\sin\left(x\right)\right)+6x\left(3^x-\cos\left(x\right)\right)$$

Answer

None of the above

Question 9

$$\frac{d}{dx}\left[\frac{\left(3x^2\sqrt[4]{x}-4\ln\left(x\right)\right)}{\ln\left(x\right)+2x^4}\right]$$

Accepted Answer

$$\frac{\left(\left(\ln\left(x\right)+2x^4\right)\left(\frac{27}{4}x^{\frac{5}{4}}-\frac{4}{x}\right)-\left(3x^{\frac{9}{4}}-4\ln\left(x\right)\right)\left(\frac{1}{x}+8x^3\right)\right)}{\left[\ln\left(x\right)+2x^4\right]^2}$$

Answer

$$\frac{\left(\left(3x^{\frac{9}{4}}-4\ln\left(x\right)\right)\left(\frac{1}{x}+8x^3\right)-\left(\ln\left(x\right)+2x^4\right)\left(\frac{27}{4}x^{\frac{5}{4}}-\frac{4}{x}\right)\right)}{\left[\ln\left(x\right)+2x^4\right]^2}$$

Answer

$$\frac{\left(\left(3x^{\frac{9}{4}}-4\ln\left(x\right)\right)\left(x+8x^3\right)-\left(\ln\left(x\right)+2x^4\right)\left(\frac{27}{4}x^{\frac{5}{4}}-4x\right)\right)}{\left[\ln\left(x\right)+2x^4\right]^2}$$

Answer

None of the above

Question 10

$$\frac{d}{dx}\left[\left(-3e^x-2^x\right)\left(-1+\frac{2x}{\cos\left(x\right)}\right)\right]$$

Accepted Answer

$$\left(-3e^x-2^x\right)\left(\frac{\left(2\cos\left(x\right)+2x\sin\left(x\right)\right)}{\cos^2\left(x\right)}\right)+\left(-3e^{x\ }-2^x\ln\left(2\right)\right)\left(-1+\frac{2x}{\cos\left(x\right)}\right)$$

Answer

$$\left(-3e^x-2^x\right)\left(-1+\frac{\left(2\cos\left(x\right)+2x\sin\left(x\right)\right)}{\cos^2\left(x\right)}\right)+\left(-3e^{x\ }-2^x\ln\left(2\right)\right)\left(-1+\frac{2x}{\cos\left(x\right)}\right)$$

Answer

$$\left(-3e^x-2^x\ln\left(2\right)\right)\left(\frac{\left(2\cos\left(x\right)+2x\sin\left(x\right)\right)}{\cos^2\left(x\right)}\right)+\left(-3e^{x\ }-2^x\right)\left(-1+\frac{2x}{\cos\left(x\right)}\right)$$

Answer

None of the above

Product & Quotient Derivative Rules

10 questions

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