# Derivative Of sin^2x, sin^2(2x) & More

Derivative Of sin^2x, sin^2(2x) – The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. Common trigonometric functions include sin(x), cos(x) and tan(x). For example, the derivative of f(x) = sin(x) is represented as f ′(a) = cos(a). f ′(a) is the rate of change of sin(x) at a particular point a.

All derivatives of circular trigonometric functions can be found using those of sin(x) and cos(x). The quotient rule is then implemented to differentiate the resulting expression. Finding the derivatives of the inverse trigonometric functions involves using implicit differentiation and the derivatives of regular trigonometric functions.

## Derivative of sin^2x

You would use the chain rule to solve this. To do that, you’ll have to determine what the “outer” function is and what the “inner” function composed in the outer function is.

In this case, sin(x) is the inner function that is composed as part of the sin²(x). To look at it another way, let’s denote u=sin(x) so that =sin²(x). Do you notice how the composite function works here? The outer function of  squares the inner function of u=sin(x). Don’t let u confuse you, it’s just to show you how one function is a composite of the other. Once you understand this, you can derive it.

So, mathematically, the chain rule is:

The derivative of a composite function F(x) is:

F'(x)=f'(g(x))(g'(x)) Or, in words:

the derivative of the outer function (with the inside function left alone!) times the derivative of the inner function.1) The derivative of the outer function(with the inside function left alone) is:

d/dx()=2u
(I’m leaving u in for now but you can sub in u=sin(x) if you want to while you’re doing the steps. Remember that these are just steps, the actual derivative of the question is shown at the bottom)2) The derivative of the inner function:

d/dx{sin(x)}=cos(x)

Combining the two steps through multiplication to get the derivative:

d/dx{sin2(x)}=2ucos(x)=2sin(x)cos(x)

## The derivative of sin^2(2x)

Here I would try to use the Chain Rule applied to three functions one nested into the other:
the first and all-embracing function is ()2; the next one is the sin function and the last one the 2x function.

I will use the Chain Rule deriving each one as if alone (regardless of the argument) and I will multiply each individual derivative together using, as visual help, a sequence of red-blue-green colors to identify each derivative:

giving:
f(x)=4sin(2x)cos(2x)

This function can be compressed (giving: 2sin(4x)) using a trigonometric identity but I do not want to confuse the procedure.

what is the derivative of sin^2x

## The derivative of ln(sin^2x)

Use the chain rule.

You can break down your function into the logarithm, the square, and the sinus function like follows:

f(u)=ln(u)
u(v)=(x)
v(x)=sin(x)

Now, you need to compute the three derivatives of those three functions (and afterward plug the respective values of u and v):

f(u)=1/u=1/=1/sin²x
u(v)=2v=2sinx
v(x)=cos(x)

Now, the only thing left to do is multiplying those three derivatives: Hope that this helped!

## The derivative of sin^2x + cos^2x

From the given
y=sin²x+cos²x

The right side of the equation is =1

y=1

dy/dx=d/dx(1)=0

or we can do it this way.

God bless I hope the explanation is useful.

## What is the derivative of sin^2x

d/dx of sin^2 x

We’ll apply chain rule here.

First, differentiate the square part i.e. ^2

which will give the answer. 2sinx

Then, differentiate the trigonometric function, i.e. sin x

which will give the answer. cos x

Then differentiate the x…

which is 1

And multiply them all, as says the chain rule,

2sinx* cosx * 1

=2sin xcos x

which can also be written as sin 2x.

Therefore, d/dx of sin^2 x= 2sin xcos x= sin2x

## What is the derivative of sin2x?

Namely, dy/dx= 2*cos(2x). Remember that the chain rule states that one “differentiates the outside, leaving the inside alone, then differentiates the inside function.” The derivative of the sin(x) with respect to x is the cos(x), and the derivative of 2x with respect to x is simply 2.

## Is sin2x the same as 2sinx?

In plain English, 1 is multiplied by whereas, 2 is the sine of 2 multiplied by x, or twice angle x. So two very different results. Interestingly as sine is a periodic (repeating) waveform, these graphs will intersect periodically and the two expressions will evaluate to the same result!

## What is the formula of sin2x?

2. Double-angle formulassin 2x = 2sin x cos x, cos 2x = cos2x − sin2x = 2cos2x − 1=1 − 2sin2x.

## What does Derivative Of sin^2x mean?

Tech Mechanical Engineering, Institute of Engineering and Management. Answered May 13, 2017. Sin x^2 is the “sine of (x-squared)”, so it is an ordinary sine function. Sin^2 x is “sine-squared of x” which is a different function from the sine function.

## How do you integrate Cos 2x?

Here’s how you integrate cos2 x:
• Use the half-angle identity for cosine to rewrite the integral in terms of cos 2x:
• Use the Constant Multiple Rule to move the denominator outside the integral:
• Distribute the function and use the Sum Rule to split it into several integrals:
• Evaluate the two integrals separately:

## How do you integrate sin^2(x)?

You cannot directly integrate sin^2(x). Use trigonometric identities and calculus substitution rules to solve the problem. Use the half-angle formula, sin^2(x) = 1/2*(1 – cos(2x)) and substitute into the integral so it becomes 1/2 times the integral of (1 – cos(2x)) dx.

## What is the chain rule in calculus?

The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x².

## What does Derivative Of sin^2x mean?

Tech Mechanical Engineering, Institute of Engineering and Management. Answered May 13, 2017. Sin x^2 is the “sine of (x-squared)”, so it is an ordinary sine function. Sin^2 x is “sine-squared of x” which is a different function from the sine function.

## What does Cos 2x mean?

Cos(2x) is just a “double angle” trig function… there’s nothing to “do” on a calculator unless we were trying to evaluate the cosine for some value of 2x where x is an angle and 2x is twice the measure of that angle.