Derivative of Tan 2x

The derivative of tan 2x is given by 2 sec²(2x) which can be calculated using different methods such as chain rule, the first principle of derivatives, and quotient rule. Differentiation of tan 2x is the process of determining the derivative of tan 2x which gives the rate of change in the trigonometric function tan 2x with respect to the angle x.

In this article, we will calculate the derivative of tan 2x using various methods which include the first principle of differentiation, chain rule, and quotient rule with the help of trigonometric formulas and identities. We will also solve some examples based on the derivative of tan 2x for a better understanding.

The derivative of tan 2x is twice the square of secant function with angle 2x, that is, 2 sec²(2x). Mathematically, the derivative of tan 2x is written as d(tan 2x)/dx = 2 sec²(2x) or (tan 2x)' = 2 sec²(2x). The process of finding the derivative of tan 2x is called the differentiation of tan 2x. Let us now see the formula for the derivative of tan 2x.

The formula for the derivative of tan 2x is:

• d(tan 2x)/dx = 2 sec²(2x)
• (tan 2x)' = 2 sec²(2x)

We know that the derivative of tan 2x is 2 sec²(2x). We will prove this using the first principle of differentiation. To derive the derivative of tan 2x, we will use the following limits and differentiation formulas:

• \(\lim_{x\rightarrow 0} \dfrac{\sin x}{x} = 1\)
• \(f'(x)=\lim_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}\)
• sin(a - b) = sin a cos b - sin b cos a
• sec x = 1/cos x
• tan x = sin x/cos x

\(\begin{align} \frac{\mathrm{d} \tan 2x}{\mathrm{d} x}&=\lim_{h\rightarrow 0}\dfrac{\tan 2(x+h)-\tan (2x)}{h}\\&=\lim_{h\rightarrow 0}\dfrac{\tan (2x+2h)-\tan (2x)}{h}\\&=\lim_{h\rightarrow 0}\dfrac{\frac{\sin(2x+2h)}{\cos(2x+2h)}-\frac{\sin 2x}{\cos 2x}}{h}\\&=\lim_{h\rightarrow 0}\dfrac{\sin (2x+2h)\cos 2x-\sin 2x\cos (2x+2h)}{h\cos (2x+2h)\cos 2x}\\&
=\lim_{h\rightarrow 0}\dfrac{\sin (2x+2h-2x)}{h\cos (2x+2h)\cos 2x}\\&=\lim_{h\rightarrow 0}\dfrac{2\sin (2h)}{2h\cos (2x+2h)\cos 2x}\\&=2\lim_{h\rightarrow 0}\dfrac{\sin (2h)}{2h}\times \lim_{h\rightarrow 0}\dfrac{1}{\cos (2x+2h)\cos 2x}\\&=2\times 1\times \dfrac{1}{\cos^2 2x}\\&=2\sec^2 2x \end{align}\)

Hence, we have derived the derivative of tan 2x using the first principle.

Next, we will evaluate the derivative of tan 2x using the chain rule. We can write the derivative of tan 2x with respect to x as a product of the derivative of tan 2x with respect to 2x and the derivative of 2x with respect to x. For the differentiation of tan 2x, we will use the following formulas:

• (f(g(x)))’ = f’(g(x)) . g’(x)
• d(tan x)/dx = sec²x
• d(ax)/dx = a, where a is a real number

Using the above formulas, we have

d(tan 2x)/dx = d(tan 2x)/d(2x) × d(2x)/dx

= sec²(2x) × 2

= 2 sec²(2x)

Thus, we have obtained the derivative of tan 2x using the chain rule.

Now that we have calculated the derivative of tan 2x using the first principle and chain rule, we will derive the derivative of tan 2x using the quotient rule. We will use the following formulas:

• Quotient Rule - (f/g)' = (f'g - fg')/g²
• tan x = sin x/cos x
• d(sin x)/dx = cos x
• d(cos x)/dx = -sin x
• sec x = 1/cos x
• cos²x + sin²x = 1

d(tan 2x)/dx = d(sin 2x/cos 2x)/dx

= [(sin 2x)' cos 2x - sin 2x (cos 2x)']/cos²2x

= [2cos 2x. cos 2x - sin x. (-2sin 2x)]/cos²2x

= 2(cos²2x + sin²2x)/cos²2x

= 2/cos²2x

= 2 sec²(2x)

Hence we have obtained the derivative of tan 2x using the quotient rule.

Important Notes on Derivative of Tan 2x

• The anti-derivative of tan 2x is (-1/2) ln|cos 2x| + C which is nothing but the integral of tan 2x.
• The derivative of tan 2x is written as d(tan 2x)/dx = 2 sec²(2x) or (tan 2x)' = 2 sec²(2x).
• Derivative of tan 2x can also be determined using the tan 2x formula.

Topics Related to Derivative of Tan 2x

• Tan 2x Formula
• Cos 2x
• Sin 3x

What is the Derivative of Tan 2x in Trigonometry?

The derivative of tan 2x is twice the square of secant function with angle 2x, that is, 2 sec²(2x).

How to Find the Derivative of Tan 2x?

The derivative of tan 2x can be calculated using various methods which include the first principle of differentiation, chain rule, and quotient rule with the help of trigonometric formulas and identities.

What is the Derivative of Tan^2(x)?

The derivative of tan²x can be calculated using the chain rule which is given by d(tan²x)/dx = 2 tan x sec²x.

What is the Derivative of Sin(tan 2x)?

The derivative of sin(tan 2x) is 2 sec²2x cos(tan 2x)

How to Find the Derivative of Tan 2x + Sec 2x?

The derivative of tan 2x + sec 2x can be calculated using the chain rule which is given by 2 sec²2x + 2 sec 2x tan 2x.