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

If $${y}=\sqrt{4 x+1}$$, evaluate $$\frac{d y}{d x}$$

Accepted Answer

Covalent bond

Answer

$$\frac{2}{\sqrt{4 x+1}}$$

Answer

$$\frac{2}{\sqrt{x}}$$

Answer

$$\frac{1}{2 \sqrt{4 x+1}}$$

Answer

$$2 \sqrt{4 x+1}$$

Question 4

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 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

Many students find Derivatives Chain Rule And Other Advanced Topics for Calculus particularly challenging. Strengthen your concepts and gain confidence for your exams by solving free questions for Derivatives Chain Rule And Other Advanced Topics from for Calculus preparation.