# How Solve it? Differentiate this function, thank you!

Mar 15, 2017

$\frac{\mathrm{df}}{\mathrm{dx}} = 20 x \left(4 {x}^{2} - 7 x - 8\right) \left(6 {x}^{2} - 7 x - 4\right)$

#### Explanation:

As we have to find derivative of a product of polynomials, we can use product rule here. It states that if $f \left(x\right) = g \left(x\right) h \left(x\right) k \left(x\right)$

then $\frac{\mathrm{df}}{\mathrm{dx}} =$

$\frac{\mathrm{dg}}{\mathrm{dx}} \times h \left(x\right) \times k \left(x\right) + \frac{\mathrm{dh}}{\mathrm{dx}} \times g \left(x\right) \times k \left(x\right) + \frac{\mathrm{dk}}{\mathrm{dx}} \times g \left(x\right) \times h \left(x\right)$

Here $f \left(x\right) = 5 {x}^{2} {\left(4 {x}^{2} - 7 x - 8\right)}^{2}$

= $5 {x}^{2} \left(4 {x}^{2} - 7 x - 8\right) \left(4 {x}^{2} - 7 x - 8\right)$

Hence $\frac{\mathrm{df}}{\mathrm{dx}} =$

$5 \times 2 x \left(4 {x}^{2} - 7 x - 8\right) \left(4 {x}^{2} - 7 x - 8\right) + \left(8 x - 7\right) \times 5 {x}^{2} \left(4 {x}^{2} - 7 x - 8\right) + \left(8 x - 7\right) \times 5 {x}^{2} \left(4 {x}^{2} - 7 x - 8\right)$

= $10 x {\left(4 {x}^{2} - 7 x - 8\right)}^{2} + 2 \times 5 {x}^{2} \left(8 x - 7\right) \left(4 {x}^{2} - 7 x - 8\right)$

= $10 x \left({\left(4 {x}^{2} - 7 x - 8\right)}^{2} + x \left(8 x - 7\right) \left(4 {x}^{2} - 7 x - 8\right)\right)$

= $10 x \left(4 {x}^{2} - 7 x - 8\right) \left(\left(4 {x}^{2} - 7 x - 8\right) + x \left(8 x - 7\right)\right)$

= $10 x \left(4 {x}^{2} - 7 x - 8\right) \left(4 {x}^{2} - 7 x - 8 + 8 {x}^{2} - 7 x\right)$

= $10 x \left(4 {x}^{2} - 7 x - 8\right) \left(12 {x}^{2} - 14 x - 8\right)$

= $20 x \left(4 {x}^{2} - 7 x - 8\right) \left(6 {x}^{2} - 7 x - 4\right)$