Question

Finding Derivatives Implicity In Exercise, find dy/dx implicity.
4xy + ln(x2y) = 7

Answer 1

To find dy/dx implicitly, differentiate both sides of the equation, applying the product rule to 4xy and the properties of logarithms and chain rule to ln(x^2y). Finally, solve for dy/dx.

Step-by-step explanation:

To find dy/dx implicitly for the equation 4xy + ln(x2y) = 7, we apply the rules of differentiation with respect to x, treating y as a function of x (y=y(x)). Here is the step-by-step process:

1. Differentiate both sides of the equation with respect to x. The left side includes the product 4xy and the natural logarithm ln(x2y).
2. For the term 4xy, use the product rule: d(uv)/dx = u'v + uv'. Here, u=4x and v=y. So, we get u'=4 and v'=dy/dx (since y is a function of x).
3. For the term ln(x2y), apply the chain rule and properties of logarithms: ln(A*B) = ln(A) + ln(B). Thus, ln(x2y) = ln(x2) + ln(y).
4. Differentiate ln(x2) to get 2/x and ln(y) with respect to x to get dy/dx divided by y.
5. Combine these derivatives and solve for dy/dx.

By combining all the derived terms, we obtain the expression for dy/dx.

Answer 2

Answer: The derivative would be
`(dy)/(dx)=((-2-4x)y)/(x(4xy+1))`

Step-by-step explanation:

Since we have given that

`4xy + ln(x2y) = 7`

We need to find the derivative implicity
`(dy)/(dx)`

so, we will assume y as constant when we derivative the function w.r.t x:

`4xy+\ln x^2+\ln y=7\\\\4+4x(dy)/(dx)+(1)/(x^2)* 2x+(1)/(y)* (dy)/(dx)=0\\4+4x(dy)/(dx)+(2)/(x)+(1)/(y)* (dy)/(dx)=0\\\\(4x+(1)/(y))(dy)/(dx)=-(2)/(x)-4\\\\(4xy+1)/(y)(dy)/(dx)=(-2-4x)/(x)\\\\(dy)/(dx)=((-2-4x)y)/(x(4xy+1))`

Hence, the derivative would be

`(dy)/(dx)=((-2-4x)y)/(x(4xy+1))`