Math Help

Implicit Differentiation

In a Nut Shell: Implicit differentiation is the process of differentiating both sides of an

equation and then solving for the derivative of the dependent variable such as dy/dx.

Example 1 of implicit differentiation:

Find dy/dx given xy = sin (x) , where y = y(x)

Step 1 Take the derivatives of both sides of the equation.

y + x dy/dx = cos (x)

Step 2 Solve for dy/dx x dy/dx = cos (x) - y

x dy/dx = cos (x) - sin (x) / x

Result: dy/dx = [ cos (x) - sin (x) / x ] / x

Example 2

Find dy/dx given e ^4y - ln (y) = 2x

Step 1 Differentiate both sides of this equation with respect to x.

4 e ^4y dy/dx - (1/y) dy/dx = 2

Step 2 Solve for dy/dx

[ 4 e ^4y - (1/y) ] dy/dx = 2

Result: dy/dx = 2 y / [ 4 y e ^4y - 1 ]

Try this one:

Find dy/dx given cos (y) - y ² = 8

Step 1 is to differentiate both sides of this equation.

Step 2 is to collect terms and solve for dy/dx

Result: dy/dx = 0 provided sin y - 2y ≠ 0