# How do you differentiate f(x) = (tan(x-7))/(e^(2x)-4) using the quotient rule?

Sep 28, 2016

The answer is $\frac{{\sec}^{2} \left(x - 7\right) \left({e}^{2 x} - 4\right) - 2 {e}^{2 x} \left(\tan \left(x - 7\right)\right)}{{e}^{2 x} - 4} ^ 2$

#### Explanation:

So, let's go ahead and plug in everything into the quotient rule:

For $f \frac{x}{g} \left(x\right)$: $\frac{d}{\mathrm{dx}} \left(f \frac{x}{g} \left(x\right)\right) = \frac{f ' \left(x\right) \cdot g \left(x\right) - g ' \left(x\right) \cdot f \left(x\right)}{g \left(x\right)} ^ 2$

So, all we need to do now is plug in $\tan \left(x - 7\right)$ as $f \left(x\right)$, and ${e}^{2 x} - 4$ as $g \left(x\right)$. This gives us:

$\frac{\frac{d}{\mathrm{dx}} \left(\tan \left(x - 7\right)\right) \cdot \left({e}^{2 x} - 4\right) - \frac{d}{\mathrm{dx}} \left({e}^{2 x} - 4\right) \cdot \tan \left(x - 7\right)}{{e}^{2 x} - 4} ^ 2$

Now, the most tedious part is done. The last trick is to calculate the two derivatives.

Firstly, we need to find the derivative of $\tan \left(x - 7\right)$. The derivative of $\tan \left(x\right)$ is ${\sec}^{2} \left(x\right)$. So, the derivative of $\tan \left(x - 7\right) = {\sec}^{2} \left(x - 7\right)$.

Note: you technically need to use the chain rule here, and multiply ${\sec}^{2} \left(x - 7\right)$ by the derivative of $x - 7$, but since the derivative of $x - 7$ is 1, we can ignore it.

Lastly, we need to find the derivative of ${e}^{2 x}$. The derivative of $\left({e}^{x}\right) = {e}^{x}$, but because of the $2 x$, the chain rule comes into play. As a result, this would play out into:

$\frac{d}{\mathrm{dx}} \left({e}^{2 x}\right) \cdot \frac{d}{\mathrm{dx}} \left(2 x\right) = {e}^{2 x} \cdot 2 = 2 {e}^{2 x}$

So, plugging those in, we have the final answer:

$\frac{{\sec}^{2} \left(x - 7\right) \left({e}^{2 x} - 4\right) - 2 {e}^{2 x} \left(\tan \left(x - 7\right)\right)}{{e}^{2 x} - 4} ^ 2$

Now, you could distribute the ${\sec}^{2} \left(x - 7\right)$, or foil out the ${\left({e}^{2 x} - 4\right)}^{2}$, but to be honest, you could just leave it as is.

Hope that helped :)