Find the derivative of sin(2x)cos(2x)

\frac{d}{dx}\left(\sin\left(2x\right)\cdot \cos\left(2x\right)\right)

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Answer

$2\left(1-2\sin\left(x\right)^2\right)^2-8\cos\left(x\right)^2\sin\left(x\right)^2$

Step by step solution

Problem

$\frac{d}{dx}\left(\sin\left(2x\right)\cdot \cos\left(2x\right)\right)$
1

Applying an identity of double-angle cosine

$\frac{d}{dx}\left(\left(1-2\sin\left(x\right)^2\right)\sin\left(2x\right)\right)$
2

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\sin\left(2x\right)$ and $g=1-2\sin\left(x\right)^2$

$\frac{d}{dx}\left(1-2\sin\left(x\right)^2\right)\sin\left(2x\right)+\left(1-2\sin\left(x\right)^2\right)\frac{d}{dx}\left(\sin\left(2x\right)\right)$
3

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{d}{dx}\left(1-2\sin\left(x\right)^2\right)\sin\left(2x\right)+\left(1-2\sin\left(x\right)^2\right)\frac{d}{dx}\left(2x\right)\cos\left(2x\right)$
4

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(1-2\sin\left(x\right)^2\right)\sin\left(2x\right)+2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)\frac{d}{dx}\left(x\right)$
5

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(1-2\sin\left(x\right)^2\right)\sin\left(2x\right)+1\cdot 2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)$
6

The derivative of a sum of two functions is the sum of the derivatives of each function

$\left(\frac{d}{dx}\left(-2\sin\left(x\right)^2\right)+\frac{d}{dx}\left(1\right)\right)\sin\left(2x\right)+1\cdot 2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)$
7

The derivative of the constant function is equal to zero

$\left(\frac{d}{dx}\left(-2\sin\left(x\right)^2\right)+0\right)\sin\left(2x\right)+1\cdot 2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)$
8

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\left(0-2\frac{d}{dx}\left(\sin\left(x\right)^2\right)\right)\sin\left(2x\right)+1\cdot 2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)$
9

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\left(0-2\cdot 2\frac{d}{dx}\left(\sin\left(x\right)\right)\sin\left(x\right)\right)\sin\left(2x\right)+1\cdot 2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)$
10

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\left(0-2\cdot 2\cos\left(x\right)\sin\left(x\right)\right)\sin\left(2x\right)+1\cdot 2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)$
11

Multiply $2$ times $1$

$\left(0-4\cos\left(x\right)\sin\left(x\right)\right)\sin\left(2x\right)+2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)$
12

$x+0=x$, where $x$ is any expression

$2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)-4\sin\left(2x\right)\cos\left(x\right)\sin\left(x\right)$
13

Using the sine double-angle identity

$2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)-4\cdot 2\cos\left(x\right)\sin\left(x\right)\cos\left(x\right)\sin\left(x\right)$
14

Multiply $2$ times $-4$

$2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)-8\cos\left(x\right)\sin\left(x\right)\cos\left(x\right)\sin\left(x\right)$
15

When multiplying exponents with same base you can add the exponents

$2\left(1-2\sin\left(x\right)^2\right)\cos\left(2x\right)-8\cos\left(x\right)^2\sin\left(x\right)^2$
16

Applying an identity of double-angle cosine

$2\left(1-2\sin\left(x\right)^2\right)\left(1-2\sin\left(x\right)^2\right)-8\cos\left(x\right)^2\sin\left(x\right)^2$
17

When multiplying exponents with same base you can add the exponents

$2\left(1-2\sin\left(x\right)^2\right)^2-8\cos\left(x\right)^2\sin\left(x\right)^2$

Answer

$2\left(1-2\sin\left(x\right)^2\right)^2-8\cos\left(x\right)^2\sin\left(x\right)^2$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.28 seconds

Views:

104