What is the derivative of f(x) = (−7 x^2 − 5)^8 * (2 x^2 − 9)^9?

Jul 29, 2015

The given function is the product of two functions and the derivative may be computed using he product rule.

Explanation:

Let, $g \left(x\right) = {\left(- 7 {x}^{2} - 5\right)}^{8}$ and $h \left(x\right) = {\left(2 {x}^{2} - 9\right)}^{9}$

Thus, $f \left(x\right) = g \left(x\right) \cdot h \left(x\right)$

Now, $\frac{\mathrm{df}}{\mathrm{dx}} = g \cdot \frac{\mathrm{dh}}{\mathrm{dx}} + h \cdot \frac{\mathrm{dg}}{\mathrm{dx}}$

Now,
$\frac{\mathrm{dh}}{\mathrm{dx}} = 9 {\left(2 {x}^{2} - 9\right)}^{9 - 1} \cdot \frac{d \left(2 {x}^{2} - 9\right)}{\mathrm{dx}}$

$= 9 {\left(2 {x}^{2} - 9\right)}^{8} \cdot \left(4 x\right)$

Also,
$\frac{\mathrm{dg}}{\mathrm{dx}} = 8 {\left(- 7 {x}^{2} - 5\right)}^{8 - 1} \cdot \frac{d \left(- 7 {x}^{2} - 5\right)}{\mathrm{dx}}$

$= 8 {\left(- 7 {x}^{2} - 5\right)}^{7} \cdot \left(- 14 x\right)$

Now do rearrange the terms. We already know $h \left(x\right) , g \left(x\right)$ and their derivatives $\frac{\mathrm{dh}}{\mathrm{dx}}$ and $\frac{\mathrm{dg}}{\mathrm{dx}}$.