Techniques of Differentiation The Product and Quotient Rules The Chain Rule Derivatives of Logarithmic and Exponential Functions Implicit Differentiation

The Product Rule The Quotient Rule

The Product Rule Ex. Derivative of Second Derivative of first

The Quotient Rule Ex. Derivative of denominator Derivative of numerator

Compute the Derivative Ex. = –10

The Chain Rule If f is a differentiable function of u and u is a differentiable function of x, then the composite f (u) is a differentiable function of x, and The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity.

Generalized Power Rule Ex.

The Chain Rule Ex.

Chain Rule in Differential Notation If y is a differentiable function of u and u is a differentiable function of x, then

Chain Rule Example Ex. Sub in for u

Differentiation of Logarithmic Functions Derivative of the Natural Logarithm Generalized Rule for Natural Logarithm Functions If u is a differentiable function, then

Examples Ex. Find the derivative of Ex. Find an equation of the tangent line to the graph of Slope: Equation:

Differentiation of Logarithmic Functions Derivative of a Logarithmic Function: Generalized Rule for Logarithm Functions If u is a differentiable function, then

Differentiation of Logarithmic Functions Ex.

Derivative of Logarithms of Absolute Values

Derivative of Logarithms of Absolute Values Ex. Ex.

Differentiation of Exponential Functions Derivative of ex: Generalized Rule for eu: If u is a differentiable function, then

Derivatives of Exponential Functions Ex. Find the derivative of Ex. Find the derivative of

Differentiation of Exponential Functions Derivative of bx: Generalized Rule for bu: If u is a differentiable function, then

Derivatives of Exponential Functions Ex. Find the derivative of

Implicit Differentiation y is explicitly a function of x. y is implicitly a function of x.

Implicit Differentiation (cont.) To differentiate the implicit case we use the chain rule where y is a function of x: Solve for

Tangent Line to Implicit Curve Ex. Find the equation of the tangent line to the curve at the point (2, 1).

Logarithmic Differentiation Ex. Use logarithmic differentiation to find the derivative of Apply ln Properties of ln Differentiate Solve