Question Video: Differentiating Trigonometric Functions Mathematics • Higher Education

If 𝑦 = sin (πœ‹/6) + 5 sin 7π‘₯ , find d𝑦/dπ‘₯.

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Video Transcript

If 𝑦 is equal to the sin of πœ‹ by six plus five times the sin of seven π‘₯, find d𝑦 by dπ‘₯.

The question gives us 𝑦 is equal to some function of π‘₯. And it wants us to find the first derivative of 𝑦 with respect to π‘₯. So we have d𝑦 by dπ‘₯ is equal to the derivative of the sin of πœ‹ 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 d𝑦 by dπ‘₯ is equal to the derivative of the sin of πœ‹ 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œ‹ 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‘Ž and 𝑛, the derivative of π‘Ž times the sin of 𝑛π‘₯ with respect to π‘₯ is equal to 𝑛 times π‘Ž times the cos of 𝑛π‘₯. 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, we’ve shown if 𝑦 is equal to the sin of πœ‹ by six plus five times the sin of seven π‘₯, then d𝑦 by dπ‘₯ is equal to 35 times the cos of seven π‘₯.

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