### Video Transcript

If 𝑦 is equal to the sin of 𝜋 by six plus five times the sin of seven 𝑥, find d𝑦 by d𝑥.

The question gives us 𝑦 is equal to some function of 𝑥. And it wants us to find the first derivative of 𝑦 with respect to 𝑥. So we have d𝑦 by d𝑥 is equal to the derivative of the sin of 𝜋 by six plus five times the sin of seven 𝑥 with respect to 𝑥. 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 d𝑦 by d𝑥 is equal to the derivative of the sin of 𝜋 by six with respect to 𝑥 plus the derivative of five times the sin of seven 𝑥 with respect to 𝑥. 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 𝑥 with respect to 𝑥.

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

Therefore, we’ve shown if 𝑦 is equal to the sin of 𝜋 by six plus five times the sin of seven 𝑥, then d𝑦 by d𝑥 is equal to 35 times the cos of seven 𝑥.