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Step-by-step Solution

Find the derivative using logarithmic differentiation method $\frac{d}{dx}\left(\cos\left(4x^2-5\right)^2\right)$

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Answer

$-16x\cos\left(4x^2-5\right)\sin\left(4x^2-5\right)$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\cos\left(4x^2-5\right)^2\right)$
1

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

$2\cos\left(4x^2-5\right)\frac{d}{dx}\left(\cos\left(4x^2-5\right)\right)$
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 $f'(x) = -\sin(x)\cdot D_x(x)$

$-2\cos\left(4x^2-5\right)\sin\left(4x^2-5\right)\frac{d}{dx}\left(4x^2-5\right)$

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Answer

$-16x\cos\left(4x^2-5\right)\sin\left(4x^2-5\right)$
$\frac{d}{dx}\left(\cos\left(4x^2-5\right)^2\right)$

Main topic:

Logarithmic differentiation

Used formulas:

6. See formulas

Time to solve it:

~ 0.51 seconds