The Chain Rule Section 3.6c

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (a) at x = 2 At x = 2:

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (b) at x = 3 At x = 3:

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (c) at x = 3 At x = 3:

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (d) at x = 2

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (e) at x = 2

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (f) at x = 2

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (g) at x = 3

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (h) at x = 2

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2 8 2 1/3 –3 3 3 –4 5 Evaluate the derivatives with respect to x of the following combinations at the given value of x. (h) at x = 2

Slopes of Parametrized Curves A parametrized curve (x(t), y(t)) is differentiable at t if x and y are differentiable at t. At a point on a differentiable parametrized curve where y is also a differentiable function of x, the derivatives dy/dt, dx/dt, and dy/dx are related by the Chain Rule: Usually, we write this in a different form… If all three derivatives exist and ,

Practice Problems Find the equation of the line tangent to the curve at the point defined by the given value of t. Find the three derivatives:

Practice Problems Find the equation of the line tangent to the curve at the point defined by the given value of t. The line passes through: And has slope: Equation of the tangent line:

Practice Problems Find the equation of the line tangent to the curve at the point defined by the given value of t. Derivatives: Point: Slope: Equation of the tangent line: