2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
3 min read•june 18, 2024
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Welcome back to AP Calculus with Fiveable! Now that you’ve mastered finding of trigonometric functions sinx and cosx, let’s cover the rest! Remembering these rules is key to simplifying your calculus journey. 🌟
💫 Derivatives of Advanced Trigonometric Functions
First, let’s have a glance at a summary table for quick reference.
Function
Derivative
Function: f(x)=tanx
f′(x)=sec2x
Function: g(x)=cotx
g′(x)=−csc2x
Function: h(x)=secx
h′(x)=secxtanx
Cosecant Function: k(x)=cscx
k′(x)=−cscxcotx
It's important to note that these are only valid for angles in radians, not degrees.
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Be comfortable using trigonometric identities to simplify expressions before finding derivatives. Specifically, tan(x)=cos(x)sin(x) and cot(x)=tan(x)1.
Derivative of tanx
The derivative of tanx is sec2x. Let’s consider an example:
f(x)=3tanx+2x2
To find the derivative of this equation, differentiate 3tanx and 2x2 individually.
Since the derivative of tanx is sec2x, the derivative of the first part is 3sec2x. The derivative of 2x2 is 4x. Hence, f′(x)=3sec2x+4x.
Derivative of cotx
The derivative of cotx is −csc2x. For example:
f(x)=5cotx+x
We again have to differentiate the two terms separately! The derivative of cotx is −csc2x, so the derivative of the first term is −5csc2x. The derivative of x is 1. Therefore, f′(x)=−5csc2x+1 or f′(x)=1−5csc2x.
Derivative of secx
The derivative of secx is secxtanx. As an example:
f(x)=2secx+3x3
Knowing the above trig derivative rule, the derivative of the first term is 2secxtanx. The derivative of 3x3 is 9x2. Thus, f′(x)=2secxtanx+9x2.
Derivative of cscx
Last but not least, the derivative of cscx is −cscxcotx. For instance:
f(x)=4cscx+7x2
The derivative of the first part is −4cscxcotx. The derivative of 7x2 is 14x. Therefore, f′(x)=−4cscxcotx+14x.
🏋️♂️ Practice Problems
Here are a couple of questions for you to get the concepts down!
❓ Advanced Trig Derivative Practice Questions
Find the derivatives for the following problems.
f(x)=2tan(x)+sec(x)
f(x)=csc(x)cot(x)
g(x)=tan2(6x)
h(x)=5cot(x)
💡 Before we reveal the answers, remember to use the , sum rule, and quotient rules.
🤔 Advanced Trig Derivative Practice Solutions
f′(x)=2sec2(x)+sec(x)tan(x)
f′(x)=−csc2(x)
g′(x)=2tan(6x)(cos2(6x)1)
h′(x)=−5csc2(x)
🔑
These questions combine your knowledge of all the derivative rules we’ve learned so far. Just in case you need to review, check these out:
Practice these rules, and you’ll soon find them as intuitive as the basic derivatives! Keep up the great work. 🌈
Key Terms to Review (10)
(-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.