If 𝑦 equals sin of eight 𝑥 squared minus four, find d𝑦 by d𝑥.
Well, this is a function of a function. We have eight 𝑥 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 𝑢 and 𝑢 equals 𝑔 of 𝑥 then d𝑦 by d𝑥 equals d𝑦 by d𝑢 multiplied by d𝑢 by d𝑥. So for our question, 𝑢 equals eight 𝑥 squared minus four. And so, 𝑦 equals sin of 𝑢.
Now, our formula for the chain rule requires us to have d𝑢 by d𝑥 and d𝑦 by d𝑢. Let’s start by differentiating 𝑢 with respect to 𝑥. We remember the power rule of differentiating. And so, d𝑢 by d𝑥 equals 16𝑥. 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 d𝑢. We recall that the derivative of sin of 𝑥 is cos of 𝑥. And so, d𝑦 by d𝑢 equals cos of 𝑢. Okay, so now, let’s apply the formula for the chain rule. We have that d𝑦 by d𝑥 equals d𝑦 by d𝑢, which is cos of 𝑢, multiplied by d𝑢 by d𝑥, which is 16𝑥. Remember that we defined 𝑢 to be eight 𝑥 squared minus four. So, we can replace 𝑢 in our answer to give us that d𝑦 by d𝑥 equals cos of eight 𝑥 squared minus four multiplied by 16𝑥, which we usually write the other way round as 16𝑥 cos of eight 𝑥 squared minus four. So, that is our final answer.