00:01
In this question, we are asked to find the derivative of the function f.
00:04
And we will start with the product rule.
00:07
By the product rule, the derivative of a product equals to u prime v plus u times v prime.
00:14
In our case, u is 4x plus 7 cube, and v equals to x squared minus 8 x plus 7 to the 4th power.
00:24
Therefore, f prime of x equals to the derivative of 4x plus 7 cubed.
00:35
Multiplied by x squared minus 8x plus 7 to the 4th plus 4x plus 7 cube multiplied by the derivative of x squared minus 8x plus 7 to the 4th.
00:56
Now let's calculate each derivative.
01:00
To calculate the derivative of 4x plus 7 cube we need to use the chain rule.
01:06
The function consists of two parts.
01:09
The cube part and the 4x plus 7 part.
01:13
First we will differentiate the cube part.
01:16
The derivative of a cube is 3 times 4x plus 7 squared by the power rule.
01:23
And then we need to multiply that by the derivative or the expression inside the cube, and that's going to be 4.
01:39
And to calculate the other derivative, we also need to use the chain rule...