find dy/dx if y=ln5x

if y=xlnx

y=x³lnx

y=logx/1+logx

y=sin²(lnx)

To find the derivative dy/dx for the given functions, we will apply the rules of differentiation, including the product rule, chain rule, and properties of logarithmic functions. Let’s break down each function step by step.

1. Function: y = ln(5x)

To differentiate y = ln(5x), we can use the chain rule. The derivative of ln(u) is (1/u) * (du/dx). Here, u = 5x.

• First, find du/dx: du/dx = 5
• Now, apply the chain rule: dy/dx = (1/(5x)) * 5 = 1/x

Thus, dy/dx = 1/x.

2. Function: y = x ln(x)

This function requires the product rule, which states that if y = u*v, then dy/dx = u'v + uv'. Here, let u = x and v = ln(x).

• Differentiate u: u' = 1
• Differentiate v: v' = 1/x
• Now apply the product rule: dy/dx = (1)(ln(x)) + (x)(1/x) = ln(x) + 1

So, dy/dx = ln(x) + 1.

3. Function: y = x^3 ln(x)

Again, we will use the product rule. Let u = x^3 and v = ln(x).

• Differentiate u: u' = 3x^2
• Differentiate v: v' = 1/x
• Applying the product rule gives us: dy/dx = (3x^2)(ln(x)) + (x^3)(1/x) = 3x^2 ln(x) + x^2

Thus, dy/dx = 3x^2 ln(x) + x^2.

4. Function: y = log(x)/(1 + log(x))

For this function, we will use the quotient rule, which states that if y = u/v, then dy/dx = (u'v - uv')/v^2. Here, let u = log(x) and v = 1 + log(x).

• Differentiate u: u' = 1/(x ln(10))
• Differentiate v: v' = 1/(x ln(10))
• Now apply the quotient rule: dy/dx = [(1/(x ln(10)))(1 + log(x)) - (log(x))(1/(x ln(10)))] / (1 + log(x))^2

After simplifying, we find that dy/dx = (1/(x ln(10))) / (1 + log(x))^2.

5. Function: y = sin^2(ln(x))

For this function, we will use the chain rule. Let u = ln(x) and y = sin^2(u).

• Differentiate y with respect to u: dy/du = 2sin(u)cos(u) = sin(2u)
• Now differentiate u: du/dx = 1/x
• Applying the chain rule gives us: dy/dx = sin(2ln(x)) * (1/x)

Thus, dy/dx = sin(2ln(x))/x.

In summary, the derivatives for the functions are:

• y = ln(5x) → dy/dx = 1/x
• y = x ln(x) → dy/dx = ln(x) + 1
• y = x^3 ln(x) → dy/dx = 3x^2 ln(x) + x^2
• y = log(x)/(1 + log(x)) → dy/dx = (1/(x ln(10))) / (1 + log(x))^2
• y = sin^2(ln(x)) → dy/dx = sin(2ln(x))/x