Derivative of Secant: Find dy/dx

In summary, the derivative of secant is equal to (secx)(tanx), and can be found using the quotient rule or chain rule. It can be simplified using trigonometric identities and has significant applications in calculus, physics, and engineering. The derivative of secant can be negative, positive, or zero depending on the value of x and the function's domain and range should be considered when interpreting its sign.
  • #1
Karol
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Homework Statement


Find dy/dx for ##~y=\sec^2(5x)##

Homework Equations


Secant and it's derivative:
$$\sec\,x=\frac{1}{\cos\,x}$$
$$\sec'\,x=\tan\,x\cdot\sec\,x$$

The Attempt at a Solution


$$y=\sec^2(5x)~\rightarrow~y'=2\cdot 5 \cdot \sec(5x)\tan(5x)\sec(5x)=10\tan(5x)\sec^2(5x)$$
The answer should be:
$$y'=-\frac{2x+1}{2}\frac{1}{\sqrt{(x^2+x-1)^3}}$$
 
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  • #2
Your solution is correct to the given problem.
 
  • #3
Thank you very much jambaugh
 

1. What is the derivative of secant?

The derivative of secant is equal to the product of the secant and tangent, or (secx)(tanx).

2. How do you find the derivative of secant?

To find the derivative of secant, you can use the quotient rule or the chain rule. The quotient rule is often easier to use and involves taking the derivative of the numerator multiplied by the denominator, minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator.

3. Can the derivative of secant be simplified?

Yes, the derivative of secant can be simplified using trigonometric identities. For example, (secx)(tanx) can be rewritten as sinx.

4. What is the significance of the derivative of secant?

The derivative of secant is important in solving problems involving rates of change and optimization in calculus. It is also used in physics and engineering to calculate velocity and acceleration.

5. Can the derivative of secant be negative?

Yes, the derivative of secant can be negative, positive, or zero depending on the value of x. It is important to consider the domain and range of the function when interpreting the sign of the derivative.

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