# Proof Derivatives of sec^2x: 2sec^3x*sinx

• gordda
In summary, the conversation is about proving the derivatives of sec^2x and determining whether the quotient rule or product rule should be used. The use of the chain rule is also discussed as a possible method. Ultimately, the conversation concludes with the use of the chain rule to prove the derivatives of sec^2x, providing a step-by-step explanation for clarification. The person thanking for the help and expressing relief in understanding the concept.
gordda
hi, new to this site.
i was given this question on a recent test and not quite sure if i got it right.
it was:

prove the derivatives of sec^2x is 2sec^3x*sinx.

i wasn't really sure of the answer. so if anyone could help out, it would be much appreciated.
thanx :)

i didn't know to use the quotient rule or the product rule first.

Do you know the chain rule? If so, it should be easy.

If not, I'd probably use the product rule first:

$$\frac{d}{dx}\sec^2{x} = \sec x \frac{d}{dx}\sec x \; + \; \sec x \frac{d}{dx}\sec x = 2 \sec x \frac{d}{dx} \sec x$$

and from there, you can rewrite it as

$$2 \sec x \frac{d}{dx} \frac{1}{\cos x}$$

and use the quotient rule.

There are many other ways to do it, of course, and you could do it by using the quotient rule first if you felt like it.

Last edited:
I get the product/ quotient rule. but with the chian rule doesn't have to be a function within a function. sec^2x isn't a function within a function is it?

Let

$$f(x) = x^2$$

$$g(x) = \sec x.$$

Then

$$\sec^2 x = f(g(x))$$

of course, you still have to differentiate $\sec$~

You can do that using the chain rule too if you want. Let

$$h(x) = \frac{1}{x}$$

and

$$u(x) = \cos x.$$

Then

$$\sec x = h(u(x)).$$

In fact, you can just write

$$\sec^2 x = f(h(u(x)))$$

and 'peel back the layers' with the chain rule, if you want to.

Last edited:
gordda said:
hi, new to this site.
i was given this question on a recent test and not quite sure if i got it right.
it was:

prove the derivatives of sec^2x is 2sec^3x*sinx.

i wasn't really sure of the answer. so if anyone could help out, it would be much appreciated.
thanx :)
Here's the "Chain Rule" method:
(d/dx){sec^2(x)} = 2*{sec(x)}*(d/dx){sec(x)} =
= 2*{sec(x)}*(d/dx){cos^(-1)(x)} =
= 2*{sec(x)}*{(-1)*cos^(-2)(x)}*(d/dx){cos(x)} =
= 2*{sec(x)}*{(-1)*cos^(-2)(x)}*{(-1)*sin(x)} =
= 2*{sec(x)}*{(-1)*sec^(2)(x)}*{(-1)*sin(x)} =
= 2*{sec^3(x)}*{sin(x)}

~~

Last edited:
i sort of get it. but i think i would use the product rule to derive the question.
thanx for the help i can now sleep at night even though i know i got the question wrong :)

## What is the proof of the derivative of sec^2x?

The derivative of sec^2x is 2sec^3x*sinx, which can be proven using the quotient rule and the chain rule. The quotient rule states that the derivative of two functions divided by each other is equal to (the derivative of the first function times the second function) minus (the first function times the derivative of the second function) divided by the square of the second function. Applying this rule to sec^2x, we get (2secx*tanx) - (sec^2x*secx*tanx) / (sec^2x)^2, which simplifies to 2sec^3x*sinx.

## What is the chain rule used in the proof of the derivative of sec^2x?

In the proof of the derivative of sec^2x, the chain rule is used to find the derivative of the inside function, which in this case is secx. The chain rule states that the derivative of a composite function is equal to the derivative of the outside function evaluated at the inside function, multiplied by the derivative of the inside function. In this case, the outside function is the power function, and the inside function is secx. Therefore, the derivative of the inside function is secx*tanx, which is used in the quotient rule to find the derivative of sec^2x.

## What is the derivative of secx?

The derivative of secx is secx*tanx. This can be derived using the quotient rule, with the first function being 1 and the second function being cosx. The derivative of 1 is 0, and the derivative of cosx is -sinx. Therefore, the derivative of secx is (1*-sinx) - (0*cosx) / cosx^2, which simplifies to secx*tanx.

## How is the derivative of sec^2x used in calculus?

The derivative of sec^2x is used in calculus to find the rate of change of a function that involves secant and tangent functions. It can also be used to find the slope of a tangent line to a curve at a specific point. Additionally, the derivative of sec^2x is a fundamental component in the integration of trigonometric functions.

## What are some common applications of the derivative of sec^2x?

The derivative of sec^2x has many applications in mathematics, physics, and engineering. It is used to calculate the velocity and acceleration of objects moving along curved paths, as well as to analyze the motion of pendulums and springs. It is also used in the fields of optics, electromagnetism, and fluid dynamics to model and solve various problems. Additionally, the derivative of sec^2x is used in the creation of computer graphics and animations.

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