Video Transcript
Given that 𝑥 is equal to 𝑦 to the
power of five plus the square root of 𝑦 plus the cube root of 𝑦 squared, find d𝑦
by d𝑥.
We can find d𝑦 by d𝑥 by using the
formula for the derivative of the inverse function. Which is that d𝑦 by d𝑥 is equal
to one over d𝑥 by d𝑦. So we start by differentiating 𝑥
with respect to 𝑦. Let’s start by rewriting some of
the terms in 𝑥. We can rewrite the square root of
𝑦 as 𝑦 to the power of a half and the cube root of 𝑦 squared as 𝑦 to the power
of two over three. Now, we can use the power rule for
differentiation in order to differentiate 𝑥 with respect to 𝑦, term by term.
We multiply by the power and
decrease the power by one. This gives us that d𝑥 by d𝑦 is
equal to five 𝑦 to the power of four plus one-half 𝑦 to the power of negative
one-half plus two-thirds 𝑦 to the power of negative one-third. We can rewrite that fractional
powers of 𝑦 back in their surd form. And then, we can combine these
three terms into one fraction by creating a common denominator of six 𝑦. Our first term becomes 30𝑦 to the
power of five over six 𝑦. Our second term becomes three
multiplied by the square root of 𝑦 over six 𝑦. And our third term becomes four
multiplied by the cube root of 𝑦 squared over six 𝑦. We obtain that d𝑥 by d𝑦 is equal
to 30 multiplied by 𝑦 to the power of five plus three multiplied by the square root
of five plus four multiplied by the cube root of 𝑦 squared all over six 𝑦.
Now, we can apply the formula for
the derivative of the inverse function. And this gives us our solution that
d𝑦 by d𝑥 is simply the reciprocal of d𝑥 by d𝑦.
