# What is the implicit derivative of #1=xy-sinxy#?

By signing up, you agree to our Terms of Service and Privacy Policy

To find the implicit derivative of the equation (1 = xy - \sin(xy)), we differentiate both sides of the equation with respect to (x), treating (y) as an implicit function of (x).

[\frac{d}{dx}(1) = \frac{d}{dx}(xy - \sin(xy))]

[0 = \frac{d}{dx}(xy) - \frac{d}{dx}(\sin(xy))]

[0 = y + x\frac{dy}{dx} - \cos(xy)\left(\frac{d(xy)}{dx}\right)]

[0 = y + x\frac{dy}{dx} - \cos(xy)(y + x\frac{dy}{dx})]

[0 = y + x\frac{dy}{dx} - y\cos(xy) - x\frac{dy}{dx}\cos(xy)]

[0 = y(1 - \cos(xy)) + x\frac{dy}{dx}(1 - \cos(xy))]

[\frac{dy}{dx} = \frac{y(1 - \cos(xy))}{x(1 - \cos(xy))}]

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the derivative of #ln(1/x)#?
- What is the derivative of #mx+b#?
- What is the derivative of #sqrt(x)/(x^3+1)# using the quotient rule?
- How do you differentiate #f(x)=x/(2x-2)-2/(2x-2)# using the quotient rule?
- Let #f(x) = x^2 - 4x - 5#, x > 2, how do you find the value of #(df^-1)/dx# (or the derivative of the inverse of f(x)), at the point x = 0 = f(5)?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7