The Chain Rule

Multiple Choice

Warm Up SAT

The UAE Hope Probe traveled more than 493 million kilometers in space over the course of seven months on its journey to Mars on February 9, 2021, making the UAE the fifth country to reach the Red Planet, celebrating the fiftieth anniversary of the establishment of the UAE according to the relationship

$$f\left(x\right)=\sqrt[4]{x}$$ , Find the slope of that track at x=1.

•   $$\frac{1}{3}$$

•   $$\frac{2}{7}$$

• $$\frac{1}{4}$$

• $$\frac{3}{11}$$

Match

Before

Match the following

• $$\frac{d}{dx}\left(100\right)=$$  

• $$\frac{d}{dx}\left(100x\right)=$$  

• $$\frac{d}{dx}\left(x^{100}\right)=$$  

• $$\frac{d}{dx}\left(\frac{1}{100}x^{100}\right)=$$  

• $$\frac{d}{dx}\left(100x^2+100x\right)=$$  

0

100

100x⁹⁹

x⁹⁹

200x+100

• 0

• 100

• 100x⁹⁹

• x⁹⁹

• 200x+100

Multiple Choice

Your Turn

$$\frac{dy}{dx}\ if\ y=\left(2x^2+4\right)^3$$  

• $$\left(12x+4\right)\left(2x^2+4\right)^2$$  

• $$3\left(2x^2+4\right)^2$$  

• $$12x\left(2x^2+4\right)^2$$  

• $$12\left(2x^2+4\right)^2$$  

Multiple Choice

Warm UP

The Dubai Metro train runs according to the function

$$f\left(x\right)=\sqrt[]{x^3+9x}$$

Find the derivative of the function?

• $$\frac{3x^2+9}{2\ \sqrt[]{x^3+9x}}$$

• $$\frac{x^2+9}{2\ \sqrt[]{x^3+9x}}$$

• $$\frac{3x^2+9x}{2\ \sqrt[]{x^3+9x}}$$

• $$\frac{3x^3+9x}{2\ \sqrt[]{x^3+9x}}$$

• $$\frac{3x^2+9}{2\ \sqrt[]{2x^3+9}}$$

Multiple Choice

Find the derivative of

$$g\left(x\right)=\left(x^2+1\right)^5$$  

• $$g'\left(x\right)=5\left(x^2+1\right)^4+2x$$  

• $$g'\left(x\right)=5\left(x^2+1\right)^4$$  

• $$g'\left(x\right)=5\left(2x\right)^4$$  

• $$g'\left(x\right)=10x\left(x^2+1\right)^4$$  

Multiple Choice

Differentiate $$\left(x+3\right)^5$$  

• $$5\left(x+3\right)^4$$  

• $$15\left(x+3\right)^4$$  

• $$5x\left(x+3\right)^4$$  

• $$15x\left(x+3\right)^4$$  

Multiple Choice

Differentiate $$\left(2x-1\right)^3$$  

• $$3\left(2x-1\right)^2$$  

• $$6\left(2x-1\right)^2$$  

• $$6x\left(2x-1\right)^2$$  

• $$2\left(2x-1\right)^3$$  

Multiple Choice

Differentiate $$\left(8-x\right)^7$$  

• $$7\left(8-x\right)^6$$  

• $$-7\left(8-x\right)^6$$  

• $$-7x\left(8-x\right)^6$$  

• $$7\left(8-x\right)^7$$  

Multiple Choice

Differentiate $$2\left(3x+4\right)^6$$  

• $$36\left(3x+4\right)^5$$  

• $$6\left(3x+4\right)^5$$  

• $$12\left(3x+4\right)^5$$  

• $$12x\left(3x+4\right)^5$$  

Multiple Choice

Differentiate $$\left(6-5x\right)^4$$  

• $$20\left(6-5x\right)^3$$  

• $$4\left(6-5x\right)^3$$  

• $$-20\left(6-5x\right)^3$$  

• $$-5\left(6-5x\right)^3$$  

Multiple Choice

Differentiate $$\sqrt[]{4x-1}$$  

• $$2(4x−1)^{-\frac{1}{2}}$$  

• $$2(4x−1)^{\frac{1}{2}}$$  

• $$4(4x−1)^{-\frac{1}{2}}$$  

• $$4(4x−1)^{\frac{1}{2}}$$  

Multiple Choice

Given  $$F\left(x\right)=\left(\frac{x}{2x+1}\right)^3$$ ,    find  $$F'\left(x\right)$$ . 

• $$3\left(\frac{x}{2x+1}\right)^2$$  

• $$3\left[\frac{1}{\left(2x+1\right)^2}\right]^2$$  

• $$\frac{1}{2}$$  

• $$\frac{3x^2}{\left(2x+1\right)^4}$$