Implicit Differentiation Section 3.7 Implicit Differentiation

Example 1 Find dy/dx if 𝑦 2 =𝑥.

Example 2 Find dy/dx if 𝑥 2 𝑦+𝑥 𝑦 2 =6.

Example 3 Find the slope of the circle 𝑥 2 + 𝑦 2 =25 at the point (3, -4).

Example 4 Show that the slope dy/dx is defined at every point on the graph of 2𝑦= 𝑥 2 +𝑠𝑖𝑛𝑦.

Example 5 Differentiate. 𝑥𝑠𝑖𝑛𝑦+𝑦𝑐𝑜𝑠𝑥=1

Implicit Differentiation Process Differentiate each side with respect to x. Collect all y’ terms on one side. Factor out y’. Solve for y’.

Example 6 Find the tangent and normal to the ellipse 𝑥 2 −𝑥𝑦+ 𝑦 2 =7 at the point (-1, 2).

Example 7 Find d²y/dx² if 2 𝑥 3 −3 𝑦 2 =8.

TOTD Find the lines that are (a) tangent and (b) normal to the curve at the given point (-1,0). 6 𝑥 2 +3𝑥𝑦+2 𝑦 2 +17𝑦−6=0

Example 8 Find the second derivative of 𝑥 2 + 𝑦 2 =1.

Example 9 Differentiate. 𝑦= 𝑥 2 3

Example 10 Differentiate. 𝑦=𝑐𝑜 𝑠 − 1 5 𝑥

TOTD Find dy/dx 𝑦=𝑥 𝑥 2 +1