Video Transcript
Find d𝑦 by d𝑥 if five 𝑦𝑒 to the
two 𝑥 equals seven 𝑒 to the five.
Now, at first glance, this does
look a little complicated. However, we can clearly see that we
can rearrange the equation to make 𝑦 the subject. We’re going to divide both sides of
the equation by five 𝑒 to the two 𝑥. On the left-hand side, that leaves
us simply with 𝑦. And on the right-hand side, we have
seven 𝑒 to the power of five over five 𝑒 to the two 𝑥. Now, actually, one over 𝑒 to the
power of two 𝑥 is equal to 𝑒 to the negative two 𝑥. So we can rewrite our equation. And we say that 𝑦 is equal to
seven 𝑒 to the power of five over five times 𝑒 to the negative two 𝑥.
Notice that seven 𝑒 to the five
over five is just a constant. So we can differentiate this using
the general formula for the derivative of the exponential function. The derivative of 𝑒 to the 𝑘𝑥
with respect to 𝑥 is 𝑘𝑒 to the 𝑘𝑥. And of course, remembering that the
constant factor rule allows us to take constants outside a derivative and
concentrate on differentiating the function of 𝑥 itself.
In other words, d𝑦 by d𝑥 is equal
to seven 𝑒 to the power of five over five times the derivative of 𝑒 to the
negative two 𝑥 with respect to 𝑥. And the derivative of 𝑒 to the
negative two 𝑥 with respect to 𝑥 is negative two 𝑒 to the negative two 𝑥. This means d𝑦 by d𝑥 is negative two
times seven 𝑒 to the power of five over five times 𝑒 to the negative two 𝑥.
Notice that the derivative can
actually be expressed in terms of 𝑦 since we said that 𝑦 was equal to seven 𝑒 to
the power of five over five times 𝑒 to the negative two 𝑥. And we can therefore say that d𝑦
by d𝑥 is equal to negative two 𝑦.
