Find d𝑦 by d𝑥, given that 𝑦
equals negative nine tan six 𝑥 minus csc seven 𝑥.
Looking at the function we’ve been
asked to differentiate, we can see that it consists of a trigonometric function, tan
of six 𝑥, and also a reciprocal trigonometric function, csc of seven 𝑥, which we
recall is equal to one over sin of seven 𝑥. In order to find each of these
derivatives then, we’re going to need to recall the standard results for
differentiating each of these functions. Firstly, the derivative with
respect to 𝑥 of tan of 𝑎𝑥 for some constant 𝑎 is equal to 𝑎 multiplied by sec
squared of 𝑎𝑥. We can see this if we recall that
tan of 𝑎𝑥 is equal to sin of 𝑎𝑥 over cos 𝑎𝑥, and then we could apply the
quotient rule to find this derivative, applying the standard results for
differentiating sine and cosine functions.
We also recall that the derivative
with respect to 𝑥 of csc of 𝑎𝑥 is equal to negative 𝑎 multiplied by csc 𝑎𝑥
multiplied by cot 𝑎𝑥. And again, we can see this if we
think of csc 𝑎𝑥 as one over sin 𝑎𝑥, and then we apply the quotient rule using
our function 𝑢 in the numerator as one and our function 𝑣 in the denominator as
sin 𝑎𝑥. Whilst it’s important to know how
to derive each of these results, it’s also really helpful to commit them to memory
because we can just quote them and then apply them in situations such as these.
Let’s find now an expression for
d𝑦 by d𝑥 then. Applying the first rule, we have
that the derivative of negative nine tan six 𝑥 is equal to negative nine multiplied
by six sec squared six 𝑥. And applying the second rule, we
have that the derivative of csc seven 𝑥 will be negative seven csc seven 𝑥 cot
seven 𝑥. So, our entire expression for d𝑦
by d𝑥 is negative nine multiplied by six sec squared six 𝑥 minus negative seven
csc seven 𝑥 cot seven 𝑥. We can simplify the coefficients to
negative 54 and positive seven and then reorder the term so that the positive term
comes first, if we wish. We find then that d𝑦 by d𝑥 is
equal to seven csc seven 𝑥 cot seven 𝑥 minus 54 sec squared six 𝑥.
Remember, we need to know how to
derive each of these results using the quotient rule, but it’s also helpful to
commit them to memory.