1
Solved example of derivatives of trigonometric functions
$\frac{d}{dx}\cos\left(3x^2+x-5\right)$
2
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 $f'(x) = -\sin(x)\cdot D_x(x)$
$-\sin\left(3x^2+x-5\right)\frac{d}{dx}\left(3x^2+x-5\right)$
3
The derivative of a sum of two functions is the sum of the derivatives of each function
$-\sin\left(3x^2+x-5\right)\left(\frac{d}{dx}\left(3x^2\right)+\frac{d}{dx}\left(x-5\right)\right)$
4
The derivative of a function multiplied by a constant ($3$) is equal to the constant times the derivative of the function
$-\sin\left(3x^2+x-5\right)\left(3\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x-5\right)\right)$
$-\sin\left(3x^2+x-5\right)\left(3\cdot 2x^{\left(2-1\right)}+\frac{d}{dx}\left(x-5\right)\right)$
Subtract the values $2$ and $-1$
$-\sin\left(3x^2+x-5\right)\left(3\cdot 2x^{1}+\frac{d}{dx}\left(x-5\right)\right)$
$-\sin\left(3x^2+x-5\right)\left(6x^{1}+\frac{d}{dx}\left(x-5\right)\right)$
Any expression to the power of $1$ is equal to that same expression
$-\sin\left(3x^2+x-5\right)\left(6x+\frac{d}{dx}\left(x-5\right)\right)$
5
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$-\sin\left(3x^2+x-5\right)\left(6x+\frac{d}{dx}\left(x-5\right)\right)$
6
The derivative of a sum of two functions is the sum of the derivatives of each function
$-\sin\left(3x^2+x-5\right)\left(6x+\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-5\right)\right)$
$-\sin\left(3x^2+x-5\right)\left(6x+\frac{d}{dx}\left(x\right)+0\right)$
$x+0=x$, where $x$ is any expression
$-\sin\left(3x^2+x-5\right)\left(6x+\frac{d}{dx}\left(x\right)\right)$
7
The derivative of the constant function ($-5$) is equal to zero
$-\sin\left(3x^2+x-5\right)\left(6x+\frac{d}{dx}\left(x\right)\right)$
8
The derivative of the linear function is equal to $1$
$-\sin\left(3x^2+x-5\right)\left(6x+1\right)$
$-\sin\left(3x^2+x-5\right)\left(6x+1\right)$