# What is the implicit derivative of #25=cosy/x-3xy#?

From the original expression we can substitute:

so:

and finally:

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To find the implicit derivative of (25 = \frac{\cos y}{x} - 3xy), we will differentiate both sides of the equation with respect to (x).

Given equation: (25 = \frac{\cos y}{x} - 3xy)

Taking the derivative of both sides with respect to (x):

(\frac{d}{dx}(25) = \frac{d}{dx}\left(\frac{\cos y}{x} - 3xy\right))

Since (25) is a constant, its derivative is (0).

Now, using the quotient rule for differentiation on (\frac{\cos y}{x}):

(\frac{d}{dx}\left(\frac{\cos y}{x}\right) = -\frac{\cos y}{x^2}\frac{dx}{dx} + \frac{x\frac{d}{dx}(\cos y)}{x^2})

Applying the chain rule to (\frac{d}{dx}(\cos y)):

(\frac{d}{dx}(\cos y) = -\sin y \cdot \frac{dy}{dx})

Substituting this back into the quotient rule:

(\frac{d}{dx}\left(\frac{\cos y}{x}\right) = -\frac{\cos y}{x^2} - \frac{x \cdot (-\sin y \cdot \frac{dy}{dx})}{x^2})

Now, differentiating (3xy) with respect to (x) using the product rule:

(\frac{d}{dx}(3xy) = 3y + 3x \frac{dy}{dx})

Now, putting it all together:

[0 = -\frac{\cos y}{x^2} - \frac{x \cdot (-\sin y \cdot \frac{dy}{dx})}{x^2} - 3y - 3x \frac{dy}{dx}]

[0 = -\frac{\cos y}{x^2} + \frac{\sin y}{x} \frac{dy}{dx} - 3y - 3x \frac{dy}{dx}]

Rearranging terms:

[\frac{\sin y}{x} \frac{dy}{dx} - 3x \frac{dy}{dx} = 3y + \frac{\cos y}{x^2}]

Factoring out ( \frac{dy}{dx} ):

[\frac{dy}{dx}(\frac{\sin y}{x} - 3x) = 3y + \frac{\cos y}{x^2}]

Finally, solving for ( \frac{dy}{dx} ):

[\frac{dy}{dx} = \frac{3y + \frac{\cos y}{x^2}}{\frac{\sin y}{x} - 3x}]

Hence, the implicit derivative of (25 = \frac{\cos y}{x} - 3xy) with respect to (x) is:

[\frac{dy}{dx} = \frac{3y + \frac{\cos y}{x^2}}{\frac{\sin y}{x} - 3x}]

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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.

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