### Video Transcript

If π¦ is equal to the sin of π by six plus five times the sin of seven π₯, find dπ¦ by dπ₯.

The question gives us π¦ is equal to some function of π₯. And it wants us to find the first derivative of π¦ with respect to π₯. So we have dπ¦ by dπ₯ is equal to the derivative of the sin of π by six plus five times the sin of seven π₯ with respect to π₯. And we can simplify this by remembering the derivative of the sum of two functions is equal to the sum of their derivatives.

Using this, we have dπ¦ by dπ₯ is equal to the derivative of the sin of π by six with respect to π₯ plus the derivative of five times the sin of seven π₯ with respect to π₯. However, we know the sin of π by six is just a constant. So its derivative is equal to zero. So we only need to evaluate the derivative of five times the sin of seven π₯ with respect to π₯.

And to differentiate this, we recall the following rule for differentiating trigonometric functions. For constants π and π, the derivative of π times the sin of ππ₯ with respect to π₯ is equal to π times π times the cos of ππ₯. Applying this, we have the derivative of five times the sin of seven π₯ with respect to π₯ is equal to seven times five times the cos of seven π₯, which we can simplify to give us 35 times the cos of seven π₯.

Therefore, weβve shown if π¦ is equal to the sin of π by six plus five times the sin of seven π₯, then dπ¦ by dπ₯ is equal to 35 times the cos of seven π₯.