Question Video: Differentiating Trigonometric Functions Using the Chain Rule

If 𝑦 = sin (8π‘₯Β² βˆ’ 4), find d𝑦/dπ‘₯.

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Video Transcript

If 𝑦 equals sin of eight π‘₯ squared minus four, find d𝑦 by dπ‘₯.

Well, this is a function of a function. We have eight π‘₯ squared minus four as the inner function and sine as the outer function. And when we want to differentiate a function of a function, we use the chain rule. The chain rule says that if 𝑦 equals 𝑓 of 𝑒 and 𝑒 equals 𝑔 of π‘₯ then d𝑦 by dπ‘₯ equals d𝑦 by d𝑒 multiplied by d𝑒 by dπ‘₯. So for our question, 𝑒 equals eight π‘₯ squared minus four. And so, 𝑦 equals sin of 𝑒.

Now, our formula for the chain rule requires us to have d𝑒 by dπ‘₯ and d𝑦 by d𝑒. Let’s start by differentiating 𝑒 with respect to π‘₯. We remember the power rule of differentiating. And so, d𝑒 by dπ‘₯ equals 16π‘₯. This is because we multiplied eight, the coefficient, by the power which is two to get 16. And then, we took one away from the power. And four is a constant which differentiates to zero.

Now, let’s find d𝑦 by d𝑒. We recall that the derivative of sin of π‘₯ is cos of π‘₯. And so, d𝑦 by d𝑒 equals cos of 𝑒. Okay, so now, let’s apply the formula for the chain rule. We have that d𝑦 by dπ‘₯ equals d𝑦 by d𝑒, which is cos of 𝑒, multiplied by d𝑒 by dπ‘₯, which is 16π‘₯. Remember that we defined 𝑒 to be eight π‘₯ squared minus four. So, we can replace 𝑒 in our answer to give us that d𝑦 by dπ‘₯ equals cos of eight π‘₯ squared minus four multiplied by 16π‘₯, which we usually write the other way round as 16π‘₯ cos of eight π‘₯ squared minus four. So, that is our final answer.

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