# What is the derivative of cos(a^3+x^3)?

Jun 20, 2016

$- \left(3 {x}^{2}\right) \left(\sin \left({a}^{3} + {x}^{3}\right)\right)$

#### Explanation:

Use chain rule for derivatives.

Consider this as $\frac{d}{\mathrm{dx}} \left(\cos \left(f \left(x\right)\right)\right)$ where $f \left(x\right)$ = ${a}^{3} + {x}^{3}$

The answer would be composition of derivatives of $\cos \left(x\right)$ (and putting x as f(x) after differentiating and f(x). Let me demonstrate this in this question.

Derivative of $\cos \left(x\right)$ is $- \sin \left(x\right)$. Now, let's substitute x with f(x)

So the answer is $\left(- \sin \left(f \left(x\right)\right) \times \left(\frac{d}{\mathrm{dx}} \left(f \left(x\right)\right)\right)\right)$.

Now, $f \left(x\right) = {a}^{3} + {x}^{3}$. Assuming a to be a constant, ${a}^{3}$ is a constant and derivative of a constant is $0$. Derivative of ${x}^{3}$ is $3 {x}^{2}$. I won't explain this because you need to learn this yourself if you can't already figure it out.

Ans: $\left(- \sin \left({a}^{3} + {x}^{3}\right) \times \left(\frac{d}{\mathrm{dx}} \left({a}^{3} + {x}^{3}\right)\right)\right)$

Final Ans: $- 3 {x}^{2} \times \sin \left({a}^{3} + {x}^{3}\right)$

Jun 20, 2016

Just another way of saying the same thing

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = - 3 {x}^{2} \sin \left({a}^{3} + {x}^{3}\right)$

#### Explanation:

Let $u = {a}^{3} + {x}^{3} \text{ "->" } \frac{\mathrm{du}}{\mathrm{dx}} = 3 {x}^{2}$

Let $y = \cos \left(u\right) \text{ "->" } \frac{\mathrm{dy}}{\mathrm{du}} = - \sin \left(u\right)$

But $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{du}}{\mathrm{dx}} \times \frac{\mathrm{dy}}{\mathrm{du}}$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = - 3 {x}^{2} \sin \left({a}^{3} + {x}^{3}\right)$

Jun 20, 2016

$\frac{d}{\mathrm{dx}} = - \sin \left({a}^{3} + {x}^{3}\right) 3 {x}^{2}$
the main function is $\cos \left(x\right)$
the sub function is ${a}^{3} + {x}^{3}$
$\frac{d}{\mathrm{dx}} = - \sin \left({a}^{3} + {x}^{3}\right) 3 {x}^{2}$