Calculus 1

Module 6:

Techniques for Differentiation

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Affirmations

• I am proactive and take the initiative to complete tasks
• I will handle whatever happens, like I always do
• I can have a positive impact on another student’s life

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Derivatives of Trigonometric Functions

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Learning Goals

1 Calculate the derivatives of sine and cosine functions, including second derivatives and beyond

2 Determine the derivatives for basic trig functions like tangent, cotangent, secant, and cosecant

Deepen your understanding and form connections within these skills:

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Derivatives of Sin x and Cos x

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Try It!

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Derivatives of Tan x, Cot x, Sec x, and Csc x

• Derivative of Tangent:
• Derivative of Cotangent:
• Derivative of Secant:
• Derivative of Cosecant:

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Try It!

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Higher-Order Derivatives of Trig Functions

Higher order functions of sinx and cosx exhibit a cyclical pattern. This pattern can be used to determine higher-order derivatives.

Steps:

1. Determine the order of the derivative you need (let’s call it n)
2. Calculate the remainder when n is divided by 4. The remainder determines the position in the cycle. The derivative:
1. Remainder 0: returns to the original function
2. Remainder 1: progresses to the next function in the cycle
3. Remainder 2: is the negative of the original function
4. Remainder 3: is the negative of the next function in the cycle

Example:

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Solution:

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The Chain Rule

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Learning Goals

Deepen your understanding and form connections within these skills:

• Explain and use the chain rule
• Use the chain rule along with other rules to differentiate functions involving powers, products, quotients, and trigonometry
• Use the chain rule to find derivatives when multiple functions are nested together

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The Chain Rule

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Try It!

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Power Rule for Composition of Functions

For all values of x for which the derivative is defined, if

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Then

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Example:

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Solution:

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Try It!

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Using the Chain Rule with Trigonometric Functions

For all values of x for which the derivative is defined,

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Chain Rule for a Composition of Three Functions

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Try It!

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Chain Rule Using Leibniz’s Notation

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Derivatives of Inverse Functions

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Learning Goals

1 Find the derivative of an inverse function

2 Identify the derivatives for inverse trig functions like arcsine, arccosine, and arctangent

Deepen your understanding and form connections within these skills:

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Inverse Function Theorem

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Try It!

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Extending the Power Rule to Rational Exponents

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Derivatives of Inverse Trigonometric Functions

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Try It!

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Implicit Differentiation

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Learning Goals

1 Use implicit differentiation to find derivatives and the equations for tangent lines

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Deepen your understanding and form connections within these skills:

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Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of functions that are not explicitly solved for one variable in terms of another. To apply implicit differentiation practically, consider finding the slope of the tangent line to the circle at a specific point.

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Problem-Solving Implicit Differentiation

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Steps:

1. Take the derivative of both sides of the equation
2. Rewrite the equation so that all terms containing dy/dx are on the left
3. Factor out dy/dx
4. Solve for dy/dx by dividing both sides of the equation by an appropriate algebraic expression

Example:

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Solution:

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Try It!

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Derivatives of Exponential and Logarithmic Functions

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Learning Goals

Deepen your understanding and form connections within these skills:

• Determine the derivatives of exponential and logarithmic functions
• Apply logarithmic differentiation to find derivatives

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Derivative of the Natural Exponential Function

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Derivative of the Natural Logarithmic Function

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Derivatives of General Exponential and Logarithmic Functions

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Try It!

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Logarithmic Differentiation

Logarithmic differentiation simplifies the process by taking the natural logarithm of both sides, which transforms multiplicative relationships into additive ones, making the derivative more straightforward to compute.

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Use Logarithmic Fifferentiation

Steps:

• Begin by logarithmizing
• Expand using logarithm properties
• Differentiate both sides
• Isolate
• Simplify the derivative

Example:

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Solution:

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Differentiation Detective: Tangent Line Scavenger Hunt

Work with your group or partner to complete the scavenger hunt problems related to trigonometric functions, the chain rule, inverse functions, implicit differentiation, and exponential and logarithmic functions.��The answer to each card will lead you to the next question.

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