1) Differentiate the following:

a) y=(x^3+5x)^6 (x^2+1)^7

(6(x^3+5x^)^5 * 3x^2+5) + (7(x^2+1)^6 *2x) * (x^3+5x)^6

b) y=(x+1)^2 / x^2

(2(x+1)*1) * (x^2) - 2x *(x+1)^2 all over (x^2)^2

Are these correct?

(6(x^3+5x^)^5 * 3x^2+5) + (7(x^2+1)^6 *2x) * (x^3+5x)^6

add ( paren
(6(x^3+5x^)^5 * (3x^2+5) + (7(x^2+1)^6 *2x) * (x^3+5x)^6

2x(x+1) all over x^4

2(x+1)/x^3

the first one can be simplified further as well, like take out (x^3+5x)^5 for starters

To differentiate the given functions, we need to apply the power rule and the quotient rule for differentiation. Let's break down each part step by step:

a) Given: y = (x^3+5x)^6 * (x^2+1)^7

To differentiate this function, we need to use the chain rule. The chain rule states that if we have a composition of two functions, f(g(x)), the derivative is given by the derivative of the outer function multiplied by the derivative of the inner function.

Let's differentiate the first part of the function: (x^3+5x)^6.

Step 1: Apply the power rule to differentiate the outer function:
Derivative of (x^3+5x)^6 = 6(x^3+5x)^5 * (3x^2+5)

Now let's differentiate the second part of the function: (x^2+1)^7.

Step 2: Apply the power rule to differentiate the outer function:
Derivative of (x^2 + 1)^7 = 7(x^2+1)^6 * (2x)

Now multiply the two derivatives obtained from steps 1 and 2:
Derivative of y = (6(x^3+5x)^5 * (3x^2+5)) + (7(x^2+1)^6 * 2x) * (x^3+5x)^6

b) Given: y = (x+1)^2 / x^2

We need to use the quotient rule here. The quotient rule states that if we have a function in the form f(x)/g(x), the derivative is given by (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2.

Let's differentiate the first part of the function: (x+1)^2.

Step 1: Apply the power rule to differentiate the numerator:
Derivative of (x+1)^2 = 2(x+1) * 1

Now, let's differentiate the denominator: x^2.

Step 2: Apply the power rule to differentiate the denominator:
Derivative of x^2 = 2x

Now, apply the quotient rule using the derivatives obtained from steps 1 and 2:
Derivative of y = [(2(x+1) * 1) * (x^2) - 2x * (x+1)^2] / (x^2)^2

Therefore, the correct differentiations are as follows:

a) y = (6(x^3+5x)^5 * (3x^2+5)) + (7(x^2+1)^6 * 2x) * (x^3+5x)^6

b) y = [(2(x+1) * 1) * (x^2) - 2x * (x+1)^2] / (x^2)^2