1
Solved example of inverse trigonometric functions differentiation
$\frac{d}{dx}\left(\arcsin\left(4x^2\right)\right)$
2
Taking the derivative of arcsine
$\frac{1}{\sqrt{1-\left(4x^2\right)^2}}\frac{d}{dx}\left(4x^2\right)$
$\frac{1}{\sqrt{1-4^2\left(x^2\right)^2}}\frac{d}{dx}\left(4x^2\right)$
Calculate the power $4^2$
$\frac{1}{\sqrt{1-1\cdot 16\left(x^2\right)^2}}\frac{d}{dx}\left(4x^2\right)$
Applying the power of a power property
$\frac{1}{\sqrt{1-16x^{4}}}\frac{d}{dx}\left(4x^2\right)$
3
The power of a product is equal to the product of it's factors raised to the same power
$\frac{1}{\sqrt{1-16x^{4}}}\frac{d}{dx}\left(4x^2\right)$
4
The derivative of a function multiplied by a constant ($4$) is equal to the constant times the derivative of the function
$4\left(\frac{1}{\sqrt{1-16x^{4}}}\right)\frac{d}{dx}\left(x^2\right)$
$4\cdot 2x^{\left(2-1\right)}\left(\frac{1}{\sqrt{1-16x^{4}}}\right)$
Subtract the values $2$ and $-1$
$4\cdot 2x^{1}\left(\frac{1}{\sqrt{1-16x^{4}}}\right)$
$8x^{1}\left(\frac{1}{\sqrt{1-16x^{4}}}\right)$
Any expression to the power of $1$ is equal to that same expression
$\frac{8x}{\sqrt{1-16x^{4}}}$
5
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{8x}{\sqrt{1-16x^{4}}}$
$\frac{8x}{\sqrt{1-16x^{4}}}$