3.5 Selecting Procedures for Calculating Derivatives

3.5 Selecting Procedures for Calculating Derivatives

Congratulations! 👏 You are now a pro at calculating derivatives! Over the past two units, you have learned many derivative rules, from the power rule to the chain rule. This knowledge has cumulated in your ability to perform advanced derivative problems involving composite, implicit, and inverse functions. So pat yourself on the back! 🤗

📝 Calculating Derivatives Practice

To effectively solve derivative problems, it‘s important to first recognize which procedures to use. Below are some practice problems to help us do such!

❓ Calculating Derivatives Practice Problems

Question 1:

Which sequence of rules can be used to differentiate $f(x) = \frac{\cos(x^{3})}{5x}$?

A) Quotient rule, then quotient rule again

B) Quotient rule, then chain rule

C) Chain rule, then chain rule again

D) Quotient rule, then product rule

Question 2:

Which sequence of rules can be used to differentiate $g(x) = 4x\cos(x)\sin(x)$?

A) Chain rule, then product rule

B) Chain rule, then chain rule again

C) Product rule, then chain rule

D) Product rule, then product rule again

Question 3:

What is the derivative of $h(x) = (3x^{3}-15x)(2x-x^{7})$?

Question 4:

What is the derivative of $f(x)= 6e^{x^3+4}$?

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✅ Answers and Solutions

Question 1: Differentiating $f(x) = \frac{\cos(x^{3})}{5x}$

Answer: B) Quotient rule, then chain rule

Explanation:

$f(x) = \frac{\cos(x^{3})}{5x}$ is the quotient of one function, $\cos(x^{3})$, and another function, $5x$. This indicates that the Quotient Rule should be used.

The function $\cos(x^{3})$ is a composite function with the outside function being $\cos(x)$ and the inside function being $x^{3}$. This means that to find its derivative, the Chain Rule needs to be applied.

The sequence of rules that should be used to differentiate $f(x)$ is B) Quotient rule, then the chain rule.

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Question 2: Differentiating $g(x) = 4x\cos(x)\sin(x)$

Answer: D) Product rule, then product rule again

Explanation:

$g(x) = 4x\cos(x)\sin(x)$ is the product of three functions, $4x$, $\cos(x)$, and $\sin(x)$. Therefore, the Product Rule needs to be applied twice to find its derivative, making D the correct answer.

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Question 3: Differentiating $h(x) = (3x^{3}-15x)(2x-x^{7})$

Answer: $-30x^{9}+120x^{7}+24x^{3}-60x$

Solution:

$h(x) = (3x^{3}-15x)(2x-x^{7})$ is the product of two functions, $3x^{3}-15x$ and $2x-x^{7}$. This means that we need to apply the Product Rule:

$\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$

The derivative of $3x^{3}-15x$ is $9x^{2}-15$, by the Power Rule and the Constant Multiple Rule.

The derivative of $2x-x^{7}$ is $2-7x^{6}$, by the Power Rule and the Constant Multiple Rule.

So, by the Product Rule, $h'(x) = (9x^{2}-15)(2x-x^{7})+(3x^{3}-15x)(2-7x^{6})$.

This simplifies to $-30x^{9}+120x^{7}+24x^{3}-60x$, which is our answer.

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Question 4: Differentiating $f(x)= 6e^{x^3+4}$

Answer: $18e^{x^3+4}x^2$

Solution:

$f(x)= 6e^{x^3+4}$ is a composite function with the outside function being $e^{x}$ and the inside function being $x^3+4$ (we can ignore the $6$ for now because we can simply multiply it at the end based on the Constant Multiple Rule).

This means that to find its derivative, we need to apply the Chain Rule:

$\frac{d}{dx}[f(g(x))]=f'(g(x)) \cdot g'(x)$

The derivative of $e^{x}$ is $e^{x}$, so $f'(g(x))$ is $e^{x^3+4}$.

The derivative of $x^3+4$ is $3x^2$.

So, by the Chain Rule, $f'(x)=6e^{x^3+4}\cdot 3x^2$, which simplifies to $18e^{x^3+4}x^2$.

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These are just a few questions to get you thinking about what rules you should be using when faced with calculating derivatives! The more you practice, the easier it’ll become. You got this! 🍀