If 𝑦 equals two sin three plus eight 𝑥, determine d𝑦 d𝑥.
So in order to determine d𝑦 d𝑥, we’re gonna have to differentiate our
function. And in order to do this, what we actually have to use is a couple of general rules to
help us out. Well first of all, we’ve got that if 𝑦 is equal to sin 𝑥, then we know that d𝑦
d𝑥 is gonna be equal to cos 𝑥.
So if we differentiate sin 𝑥 we, get cos 𝑥. Okay? This is really useful cause it helps us understand what’s gonna to happen to our sin
𝑥. However, if we look at our function, it’s in a slightly different form. Our function is more in a form like this, so we’ve got that 𝑦 is equal to 𝑎 sin 𝑓
Well because if we look at our function, what we’ve got is 𝑎, so we’ve got a
constant, so it’s two, and we got sin, and then we’ve got a function, so we’ve got
three plus eight 𝑥. So if we have it in this form, we can say that d𝑦 d𝑥 gonna be equal to 𝑎
multiplied by the derivative of the function inside our function multiplied by the
cosine of that function. Okay, great.
So now that we’ve got this, we can go on and differentiate 𝑦 equals two sin three
plus eight 𝑥. So we’ve got 𝑦 equals two sin three plus eight 𝑥. So therefore, if we now apply the rule that we looked at which was how we worked out
what d𝑦 d𝑥 would be for function in this form. And actually, by the way, this actually comes from the chain rule, so we’ve actually
used the chain rule to get this.
We’re gonna get d𝑦 d𝑥 is equal to two multiplied by derivative of three plus
eight 𝑥 with respect to 𝑥 multiplied by the cosine of three plus eight 𝑥, which
is gonna be equal to two multiplied by eight multiplied by the cosine of three plus
eight 𝑥. And we actually got this because if we differentiate with respect to 𝑥 three plus
eight 𝑥, what we’re actually gonna get is zero plus eight, which is just eight.
That’s because if we differentiate an integer, we just get zero. And if we differentiate eight 𝑥, we got eight multiplied by the exponent. So that’s gonna be eight multiplied by one, so that just gives us eight. And then it’s 𝑥 to the power of — then you subtract one from the exponent, so one
minus one, which would be 𝑥 to the power of zero, which would just give us one, so
we’re just left with eight.
Okay, great! so one more step, and then we can find our d𝑦 d𝑥. So therefore, if we multiply two by eight, we get 16. So we can say that if 𝑦 equals two sin three plus eight 𝑥, d𝑦 d𝑥 is gonna be
equal to 16 cos three plus eight 𝑥.