Introduction to the Derivative of tan 2x
Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of -sin x can be calculated by following the rules of differentiation. Or, we can directly find the derivative of tan by applying the first principle of differentiation. In this article, you will learn what the tan2x derivative is and how to calculate the tan2x differentiation by using different approaches.
What is the derivative of tan2x?
The derivative of tan 2x is a mathematical concept that measures the rate of change of the tangent function with respect to the variable x. When we differentiate tan(2x) using the chain rule, we get the result as the square of the secant function, which is a fundamental concept in trigonometry. This derivative is represented as d/dx(tan(2x)) and can be calculated by squaring the secant function. The tangent function is commonly used to determine the slope of a line at a point of change in a function. In trigonometry, it is defined as the ratio of the sine and cosine functions and is represented as:
tan x = sin x/ cos x
Differentiation of tan 2x formula
The formula for the derivative of tan(2x) is;
d/dx(tan 2x)=2sec²(2x)
This formula can be derived by applying the chain rule to the tangent function, where sec²(2x) represents the square of the secant function. This formula is an essential concept in calculus and trigonometry, and understanding it can help solve complex problems related to these fields.
How do you prove the tan2x derivative?
There are different ways to derive derivatives of tan (2x). Therefore, we can prove the tan2x differentiation by using;
- First Principle
- Chain Rule
- Quotient Rule
Each derivative rule provides a different way to compute the tan(2x) differentiation. By using these methods, we can mathematically prove the formula for finding the differential of tan2x.
Derivative of tan 2x by first principle
According to the first principle of derivative, the tan2x differentiation is equal to 2sec^2x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,
f(x)=lim f(x+h)-f(x) / h
This formula allows us to determine the rate of change of a function at a specific point by using limit definition of derivative.
Proof of derivative of tan(2x) by first principle
To prove the derivative of tan by using the first principle, replace f(x) by tan (2x) or replace by tan(3x) to calculate the derivative of tan3x by first principle. f(x)=limh➜0f(x+h)-f(x)/h
f(x) = lim tan 2(x+h) - tan (2x)/h
Therefore,
f(x) = lim [tan 2(x+h) - tan (2x)]/h
Now, by the trigonometric formula, tan x = sin x/cos x. So,
f(x) = lim [sin 2(x+h)/cos 2(x+h) - sin (2x)/cos (2x)]/h
f'(x) = lim [sin 2(x+h) cos 2x - cos 2(x+h) sin 2x]/hcos 2(x+h) cos 2x
Since sin(x+y) = sin x cos y - cos x sin y
It is the expansion of the sin x function. Therefore, we can write the above equation as;
f'(x) = lim sin 2(x+h - x)/hcos 2(x+h) cos 2x
f'(x) = lim sin 2h/h cos 2(x+h)cos 2x
More simplification,
f'(x) = lim [1/cos 2(x+h)cos 2x][sin 2h/2h]
As we know,
Lim (sin 2x/2x) = 2, we get
f'(x) = 2/cos 2x. cos 2x
Hence, the tan 2x differentiation is,
f'(x) = 2sec2 (2x)
Derivative of tan2x by chain rule
To calculate the tan 2x differentiation, we can use the chain rule since the cosine function can be expressed as a combination of two functions. The chain rule of derivatives states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. The chain rule of derivatives is defined as;
dy / dx = dy / du x du / dx
Proof of tan2x differentiation by chain rule
To prove the tan2x derivative by using chain rule, assume that tan (2x) can be written as the combination of two functions. Using this let us find the derivative of tan 2x.
y = tan u where u = 2x
Using chain rule,
y = sec2 u.du/dx
and
du/dx = 2
Now, using the value of u.
y = 2sec2 (2x)
Thus, we have derived the formula of the derivative of tan2x. You can also use dy/dt calculator to find the tan2x derivative with just a simple click.
Differentiation of tan2x using quotient rule
Another method for finding the differential of -sin x is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the secant function is the reciprocal of cosine, the derivative of cosecant can also be calculated using the quotient rule. The derivative quotient rule is defined as:
d/dx (f/g) = f(x). g(x) -g(x).f(x) /{g(x)}2
Proof of differentiation of tan 2x by quotient rule
To prove the tan 2x derivative, we can start by writing it,
f(x) = tan (2x) = sin (2x)/cos (2x) =u/v
Supposing that u = sin (2x) and v = cos (2x). Now by quotient rule,
f(x) = (vu - uv)/v2
f(x) = [cos (2x) d/dx(sin (2x)) - sin (2x).d/dx(cos (2x))] / (cos 2x)2
= [cos (2x)(2cos (2x)) + sin (2x) (2sin (2x))] / cos22x
= 2[cos2(2x) + sin2 (2x)]/ cos22x
= 2sec2 (2x)
Hence, we have derived the derivative tan 2x using the quotient rule formula calculator.
How to differentiate tan2x with a calculator?
The easiest way to calculate the derivative of tan2x is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.
- Write the function as tan(2x)in the enter function box. In this step, you need to provide input value as a function as you have to calculate the differentiation of tan2x.
- Now, select the variable by which you want to differentiate tan 2x. Here you have to choose x.
- Select how many times you want to differentiate tan2x. In this step, you can choose 2 for second, 3 for triple differentiation and so on.
- Click on the calculate button.
After completing these steps, you will receive the differentiation of tan2x within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.
Frequently asked questions
What does tan 2x mean?
Tan x is a trigonometric function which is the ratio of sine and cosine function. When the angles of sine and cosine are doubled, we obtain tan 2x.
What is tan 2x integral?
The integral of tan 2x is given by;
∫tan (2x)dx = ∫sin (2x)/cos (2x) dx
∫tan (2x)dx = (-1/2)ln|cos 2x| +c
What is the derivative of 2x2?
The derivative of 2x square, that is, 2x2 is determined using the power rule of derivatives. We have [d/dx(2x2)] = 4x. Therefore, the derivative of 2x2 is equal to 4x.