2 min read•february 15, 2024
Welcome back to AP Calculus with Fiveable! Now that you’ve mastered finding derivatives of trigonometric functions and , let’s cover the rest! Remembering these rules is key to simplifying your calculus journey. 🌟
First, let’s have a glance at a summary table for quick reference.
It's important to note that these are only valid for angles in radians, not degrees.
The derivative of is . Let’s consider an example:
To find the derivative of this equation, differentiate and individually.
Since the derivative of is , the derivative of the first part is . The derivative of is . Hence, .
The derivative of is . For example:
We again have to differentiate the two terms separately! The derivative of is , so the derivative of the first term is . The derivative of is . Therefore, or .
The derivative of is . As an example:
Knowing the above trig derivative rule, the derivative of the first term is . The derivative of is . Thus, .
Last but not least, the derivative of is . For instance:
The derivative of the first part is . The derivative of is . Therefore, .
Here are a couple of questions for you to get the concepts down!
Find the derivatives for the following problems.
💡 Before we reveal the answers, remember to use the chain rule, sum rule, and quotient rules.
Practice these rules, and you’ll soon find them as intuitive as the basic derivatives! Keep up the great work. 🌈
(-csc(x)cot(x))
: The term (-csc(x)cot(x)) represents the product of the cosecant and cotangent functions of an angle x, with a negative sign in front. It is commonly used in trigonometric identities and calculations.(-csc^2(x))
: The term (-csc^2(x)) represents the derivative of the cosecant function squared. It measures how fast the cosecant function is changing at a specific point on its graph, multiplied by -1.(cot(x))'
: The term (cot(x))' represents the derivative of the cotangent function. It measures how fast the cotangent function is changing at a specific point on its graph.(sec^2(x))
: The term (sec^2(x)) represents the derivative of the secant function. It measures how fast the secant function is changing at a specific point on its graph.Chain Rule
: The chain rule is a formula used to find the derivative of a composition of two or more functions. It states that the derivative of a composite function is equal to the derivative of the outermost function times the derivative of the innermost function.Cotangent
: Cotangent is one of six trigonometric ratios used in right triangles. It represents the ratio between adjacent side length and opposite side length in relation to an acute angle.Derivatives
: Derivatives are the rates at which quantities change. They measure how a function behaves as its input (x-value) changes.sec(x)'
: The derivative of sec(x), which represents the rate of change of the secant function with respect to x.Secant
: The secant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the adjacent side.Tangent
: A tangent line is a straight line that touches a curve at only one point and has the same slope as the curve at that point.2 min read•february 15, 2024
Welcome back to AP Calculus with Fiveable! Now that you’ve mastered finding derivatives of trigonometric functions and , let’s cover the rest! Remembering these rules is key to simplifying your calculus journey. 🌟
First, let’s have a glance at a summary table for quick reference.
It's important to note that these are only valid for angles in radians, not degrees.
The derivative of is . Let’s consider an example:
To find the derivative of this equation, differentiate and individually.
Since the derivative of is , the derivative of the first part is . The derivative of is . Hence, .
The derivative of is . For example:
We again have to differentiate the two terms separately! The derivative of is , so the derivative of the first term is . The derivative of is . Therefore, or .
The derivative of is . As an example:
Knowing the above trig derivative rule, the derivative of the first term is . The derivative of is . Thus, .
Last but not least, the derivative of is . For instance:
The derivative of the first part is . The derivative of is . Therefore, .
Here are a couple of questions for you to get the concepts down!
Find the derivatives for the following problems.
💡 Before we reveal the answers, remember to use the chain rule, sum rule, and quotient rules.
Practice these rules, and you’ll soon find them as intuitive as the basic derivatives! Keep up the great work. 🌈
(-csc(x)cot(x))
: The term (-csc(x)cot(x)) represents the product of the cosecant and cotangent functions of an angle x, with a negative sign in front. It is commonly used in trigonometric identities and calculations.(-csc^2(x))
: The term (-csc^2(x)) represents the derivative of the cosecant function squared. It measures how fast the cosecant function is changing at a specific point on its graph, multiplied by -1.(cot(x))'
: The term (cot(x))' represents the derivative of the cotangent function. It measures how fast the cotangent function is changing at a specific point on its graph.(sec^2(x))
: The term (sec^2(x)) represents the derivative of the secant function. It measures how fast the secant function is changing at a specific point on its graph.Chain Rule
: The chain rule is a formula used to find the derivative of a composition of two or more functions. It states that the derivative of a composite function is equal to the derivative of the outermost function times the derivative of the innermost function.Cotangent
: Cotangent is one of six trigonometric ratios used in right triangles. It represents the ratio between adjacent side length and opposite side length in relation to an acute angle.Derivatives
: Derivatives are the rates at which quantities change. They measure how a function behaves as its input (x-value) changes.sec(x)'
: The derivative of sec(x), which represents the rate of change of the secant function with respect to x.Secant
: The secant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the adjacent side.Tangent
: A tangent line is a straight line that touches a curve at only one point and has the same slope as the curve at that point.© 2024 Fiveable Inc. All rights reserved.
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