# How do you differentiate f(x)=sinx-4cos(5x)?

$f ' \left(x\right) = \setminus \cos x + 20 \setminus \sin \left(5 x\right)$

#### Explanation:

The given function:

$f \left(x\right) = \setminus \sin x - 4 \setminus \cos \left(5 x\right)$

Differentiating above function w.r.t. $x$ using chain rule as follows

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{d}{\mathrm{dx}} \left(\setminus \sin x - 4 \setminus \cos \left(5 x\right)\right)$

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(\setminus \sin x\right) - 4 \setminus \frac{d}{\mathrm{dx}} \left(\cos \left(5 x\right)\right)$

$f = \cos x - 4 \left(- \sin \left(5 x\right)\right) \frac{d}{\mathrm{dx}} \left(5 x\right)$

$= \setminus \cos x + 4 \setminus \sin \left(5 x\right) \left(5\right)$

$= \setminus \cos x + 20 \setminus \sin \left(5 x\right)$

Jul 23, 2018

$\cos \left(x\right) + 20 \sin \left(5 x\right)$

#### Explanation:

$f \left(x\right) = \sin \left(x\right) - 4 \cos \left(5 x\right)$

So

$f ' \left(x\right) = \cos \left(x\right) - 4 \left(- \sin \left(5 x\right) \cdot 5\right)$

$f ' \left(x\right) = \cos \left(x\right) + 20 \sin \left(5 x\right)$

Hope it helps!

Jul 30, 2018

$\cos x + 20 \sin \left(5 x\right)$

#### Explanation:

We essentially have the following:

$\textcolor{s t e e l b l u e}{\frac{d}{\mathrm{dx}} \sin x} - \textcolor{p u r p \le}{\frac{d}{\mathrm{dx}} 4 \cos \left(5 x\right)}$

What I have in blue evaluates to $\cos x$. We now have

$\cos x - 4 \frac{d}{\mathrm{dx}} \textcolor{p u r p \le}{\cos \left(5 x\right)}$

What I have in purple is a composite function with $f \left(x\right) = \cos x$ and $g \left(x\right) = 5 x$. We can find the derivative with the Chain Rule

$f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

We know:

$f \left(x\right) = \cos x \implies f ' \left(x\right) = - \sin x$
$g \left(x\right) = 5 x \implies g ' \left(x\right) = 5$

We can plug this into the Chain Rule to get

$\textcolor{p u r p \le}{- 5 \sin \left(5 x\right)}$

We now have the following:

cosx-4color(purple)((-5sin(5x)), which simplifies to

$\cos x + 20 \sin \left(5 x\right)$

Hope this helps!