Find the derivative of (sin(pix)+cos(pix))^2=2

\frac{d}{dx}\left(\left(\sin\left(\pi x\right)+\cos\left(\pi x\right)\right)^2=2\right)

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$2\left(\pi\cos\left(\pix\right)-\pi\sin\left(\pix\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$

Step by step solution

Problem

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

Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

$\frac{d}{dx}\left(\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)^2\right)=\frac{d}{dx}\left(2\right)$
2

The derivative of the constant function is equal to zero

$\frac{d}{dx}\left(\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)^2\right)=0$
3

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

$2\frac{d}{dx}\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$
4

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

$2\left(\frac{d}{dx}\left(\cos\left(\pix\right)\right)+\frac{d}{dx}\left(\sin\left(\pix\right)\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$
5

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$2\left(\frac{d}{dx}\left(\cos\left(\pix\right)\right)+\frac{d}{dx}\left(\pix\right)\cos\left(\pix\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$
6

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

$2\left(\frac{d}{dx}\left(\cos\left(\pix\right)\right)+\pi\cos\left(\pix\right)\frac{d}{dx}\left(x\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$
7

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

$2\left(\frac{d}{dx}\left(\cos\left(\pix\right)\right)+1\cdot \pi\cos\left(\pix\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$
8

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

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

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

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

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

$2\left(1\cdot \pi\cos\left(\pix\right)-1\cdot 1\cdot \pi\sin\left(\pix\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$
11

Multiply $\pi$ times $1$

$2\left(\pi\cos\left(\pix\right)-\pi\sin\left(\pix\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$

$2\left(\pi\cos\left(\pix\right)-\pi\sin\left(\pix\right)\right)\left(\cos\left(\pix\right)+\sin\left(\pix\right)\right)=0$

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Main topic:

Differential calculus

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