Quiz about Derivatives Chain Rule And Other Advanced Topics

Question 1

Evaluate $$\frac{d}{d x}(\sqrt{8 x^{2}+2 x-3})$$

Accepted Answer

Covalent bond

Answer

$$\frac{8 x+1}{\sqrt{8 x^{2}+2 x-3}}$$

Answer

$$ \frac{1}{2 \sqrt{16 x+2}}$$

Answer

$$ \frac{1}{\sqrt{16 x+2}}$$

Answer

$$\frac{8 x+1}{\sqrt{x}}$$

Question 2

Evaluate $$\frac{d}{d x}(\cos (e^{x}))$$

Accepted Answer

Covalent bond

Answer

$${-e^{x} \sin (e^{x})}$$

Answer

$$ \sin (e^{x})$$

Answer

$$ \cos (e^{x})$$

Answer

$$4x \cos (e^{x-1})$$

Question 3

Yasemin tried to find the derivative of $$x^{6} \cdot \cos (x)$$. Here is her work:

Step 1: This is the product of $$\cos (x)$$ and $$x^{6}$$. So, we should use the product rule.
Step 2:

$$\frac{d}{d x}[x^{6} \cdot \cos (x)]=\frac{d}{d x}[\cos (x)] \cdot x^{6}+\cos (x) \cdot \frac{d}{d x}[x^{6}]$$

Step 3: Finding the derivatives of the factors:

$$\frac{d}{d x}[\cos (x)] =-\sin (x) $$

$$\frac{d}{d x}[x^{6}] =6 x^{5}$$

Step 4: Putting it all together:

$$\frac{d}{d x}[x^{6} \cdot \cos (x)]=\frac{d}{d x}[\cos (x)] \cdot x^{6}+\cos (x) \cdot \frac{d}{d x}[x^{6}]=-\sin ({x}) \cdot x^{6}+6 x^{5} \cos (x)$$

Is Yasemin's work correct? If not, what's her mistake?

Accepted Answer

Covalent bond

Answer

Yasemin's work is correct.

Answer

Step 1 is incorrect. Yasemin should have used a different rule and not the product rule.

Answer

Step 2 is incorrect. Yasemin didn't state the correct product rule.

Answer

Step 3 is incorrect. Yasemin didnot differentiate $$ \cos ({x})$$ correctly.

Question 4

Wang Fang tried to find the derivative of $$\frac{x^{5}}{3 x+9}$$. Here is her work:

Step 1: This is the quotient of $$x^{5}$$ and $$3 x+9$$. So we should use the quotient rule.

Step 2:

$$\frac{d}{d x}[\frac{x^{5}}{3 x+9}]=\frac{\frac{d}{d x}[x^{5}]}{\frac{d}{d x}[3 x+9]}$$

Step 3: Finding the derivatives of the factors:

$$\frac{d}{d x}[x^{5}]=5 x^{4}$$

$$\frac{d}{d x}[3 x+9]=3$$

Step 4: Putting it all together:

$$\frac{d}{d x}[\frac{x^{5}}{3 x+9}]=\frac{\frac{d}{d x}[x^{5}]}{\frac{d}{d x}[3 x+9]}=\frac{5 x^{4}}{3}$$

Is Wang Fang's work correct? If not, what's her mistake?

Accepted Answer

Covalent bond

Answer

Step 2 is incorrect. He did not state the correct quotient rule.

Answer

Step 1 is incorrect. He should have used a different rule and not the quotient rule.

Answer

His work is correct.

Answer

Step 3 is incorrect. He did not difterentiate $$3 x+9$$ correctly.

Question 5

Ali tried to find the derivative of $$(5 x-6)^{7}$$. Here is his work:

Step 1: This is a composite function. The inner function is $${u}({x})=5 {x}-6$$ and the outer function is $${w}({x})=x^{7}$$

Step 2: We should use the chain rule:

$$\frac{d}{d x}[w(u(x))]=w^{\prime}(u(x)) \cdot u^{\prime}(x)$$

Step 3: Finding the derivatives of the inner and outer functions:

$$w^{\prime}(x)=7 x^{6} 	ext { and } u^{\prime}(x)=5$$

Step 4: Putting it all together:

$$\frac{d}{d x}[(5 x-6)^{7}]=7(5 x-6)^{6} \cdot(5) $$

$$=35(5 x-6)^{6}$$

Is Ali's work correct? If not, what's his mistake?

Accepted Answer

Covalent bond

Answer

Ali's work is correct.

Answer

step 1 is incorrect. Ali had a mistake identifying the inner and the outer functions.

Answer

step 2 is incorrect. Ali did not state the correct chain rule.

Answer

step 3 is incorrect.Ali did not differentiate $$x^{7}$$ correctly.

Derivatives Chain Rule And Other Advanced Topics Calculus Maths Problems and Solutions

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