Let's find the derivatives for each of these functions using the appropriate differentiation rules, such as the chain rule and quotient rule.

Problem 61

Given: $y = (7x + 3)^4$

To differentiate, use the chain rule:

1. Let $u = 7x + 3$, then $y = u^4$.
2. The derivative of $y$ with respect to $u$ is $\frac{dy}{du} = 4u^3$.
3. The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 7$.

Using the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 4(7x + 3)^3 \cdot 7 = 28(7x + 3)^3$

Problem 62

Given: $y = (x^2 - 6)^3$

Use the chain rule here as well:

1. Let $u = x^2 - 6$, then $y = u^3$.
2. The derivative of $y$ with respect to $u$ is $\frac{dy}{du} = 3u^2$.
3. The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 2x$.

Using the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3(x^2 - 6)^2 \cdot 2x = 6x(x^2 - 6)^2$

Problem 63

Given: $y = \frac{1}{(x^2 + 5)^3}$

Rewrite the function as: $y = (x^2 + 5)^{-3}$

Differentiate using the chain rule:

1. Let $u = x^2 + 5$, then $y = u^{-3}$.
2. The derivative of $y$ with respect to $u$ is $\frac{dy}{du} = -3u^{-4}$.
3. The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 2x$.

Using the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -3(x^2 + 5)^{-4} \cdot 2x = -\frac{6x}{(x^2 + 5)^4}$

Problem 64

Given: $f(x) = \frac{1}{(5x + 1)^2}$

Rewrite the function as: $f(x) = (5x + 1)^{-2}$

Differentiate using the chain rule:

1. Let $u = 5x + 1$, then $f(x) = u^{-2}$.
2. The derivative of $f$ with respect to $u$ is $\frac{df}{du} = -2u^{-3}$.
3. The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 5$.

Using the chain rule: $\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = -2(5x + 1)^{-3} \cdot 5 = -\frac{10}{(5x + 1)^3}$

Would you like further details or explanations on any specific step?

Related Questions:

1. How can the product rule be combined with the chain rule in differentiation?
2. What are the applications of derivatives in real-world problems?
3. How does implicit differentiation differ from the chain rule?
4. Can you differentiate composite functions that involve trigonometric terms?
5. How do we handle higher-order derivatives for similar types of functions?

Tip:

Always simplify the expression of the derivative as much as possible to make it easier for further calculations or applications.