4.2 – Implicit Differentiation

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4.2 – Implicit Differentiation Ch. 4 – More Derivatives 4.2 – Implicit Differentiation

Ex: Find the derivative of x2 + y2 = 25. This function isn’t solved for y, so we will use IMPLICIT DIFFERENTIATION! Implicit Differentiation: Differentiating when the dependent variable is not solved for. Follow these steps: Take the derivatives term by term. Use normal derivative rules for all variables and numbers. x2 and y2 are variables, but 25 is a constant… When you take the derivative of a y, include a dy/dx (or a y’) afterward. Factor out and solve for dy/dx!

Ex: Find the derivative of 4x2 + 2xy = y2. Take the derivatives term by term. Use normal derivative rules for all variables and numbers. Don’t forget about product rule for the 2xy term! When you take the derivative of a y, include a dy/dx (or a y’) afterward. Factor out and solve for dy/dx! Get all the dy/dx’s to one side and factor!

Ex: Find the slope of the line tangent to 4x2 y – 6y = x3 + 2 at (2, 1). I’ll use y’ instead of dy/dx for this problem just to show you that they are interchangeable: Don’t forget about product rule for 4x2 y! Now evaluate at the point (2, 1)…

Ex: Differentiate x = cosy. Ex: Differentiate (2x + y2)3 – y = 8. Use chain rule!

Ex: Differentiate .

Ex: Find the 2nd derivative of x3 +y2 = 9 . Find the 1st derivative and solve for y’… Now take the 2nd derivative (remember, we still include a y’ for derivatives of y)… Lastly, substitute in your previously discovered y’…

Ex: Find the 2nd derivative of 2y3 – 6x2 = y. Find the 1st derivative and solve for y’… Now take the 2nd derivative (remember, we still include a y’ for derivatives of y)… Lastly, substitute in your previously discovered y’…