Question Video: Differentiating Functions Involving Trigonometric Ratios Using the Product Rule | Nagwa Question Video: Differentiating Functions Involving Trigonometric Ratios Using the Product Rule | Nagwa

Question Video: Differentiating Functions Involving Trigonometric Ratios Using the Product Rule Mathematics • Second Year of Secondary School

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If 𝑦 = π‘₯⁡ sin 5π‘₯, determine d𝑦/dπ‘₯.

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Video Transcript

If 𝑦 is equal to π‘₯ to the power of five sin five π‘₯, determine d𝑦 by dπ‘₯.

Here, we have a function which is the product of two differentiable functions. We can, therefore, use the product rule to help us evaluate d𝑦 by dπ‘₯. This says that, for our function 𝑦 is equal to 𝑒 times 𝑣, the derivative of 𝑦 with respect to π‘₯ is equal to 𝑒 times d𝑣 by dπ‘₯ plus 𝑣 times d𝑒 by dπ‘₯. We’re going to let 𝑒 be equal to π‘₯ to the power of five and 𝑣 be equal to sin five π‘₯. We begin by evaluating the derivative of 𝑒 with respect to π‘₯. It’s five π‘₯ to the power of four. Similarly, since the derivative of sin π‘Žπ‘₯ is π‘Ž cos π‘Žπ‘₯, d𝑣 by dπ‘₯ is equal to five cos five π‘₯.

Then we substitute into the formula. 𝑒 times d𝑣 by dπ‘₯ is π‘₯ to the power of five times five cos five π‘₯. And 𝑣 times d𝑒 by dπ‘₯ is sin five π‘₯ times five π‘₯ to the power of four. And so d𝑦 by dπ‘₯, in this case, is five π‘₯ to the power of five times cos five π‘₯ plus five π‘₯ to the power of four times sin five π‘₯.

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