Question Video: Finding the First Derivative of a Function Using the Power Rule Mathematics • Higher Education

Video Transcript

Given that 𝑦 is equal to three times 𝑥 raised to the power seven times 𝑒 raised to the negative six, determine d𝑦 by d𝑥.

We’re given that 𝑦 is equal to three times 𝑥 raised to the seventh power times 𝑒 to the negative six, where 𝑒 is Euler’s number, which to five decimal places is 2.71828. And we’re asked to find the derivative of 𝑦 with respect to 𝑥.

The first thing to note is that 𝑦 is a function of 𝑥 is not as complicated as it looks. In fact, since both three and 𝑒 to the negative six are constants, if we rearrange a little so that our constants are together, we have an expression of the form 𝑦 is equal to 𝑎 times 𝑥 raised to the 𝑛th power. In our case, 𝑎 is equal to three times 𝑒 raised to the power negative six and 𝑛 is equal to seven.

We’re asked to determine d𝑦 by d𝑥. And for a function of this type, we can use the power rule for differentiation. This says that if 𝑦 is equal to 𝑎 times 𝑥 raised to the power 𝑛, d𝑦 by d𝑥 is equal to 𝑛 times 𝑎 times 𝑥 raised to the power 𝑛 minus one. That is, we multiply by the exponent 𝑛 and subtract one from the exponent to get 𝑛 minus one as our new exponent.

And remembering that, in our case, 𝑦 is equal to three times 𝑒 raised to the power negative six times 𝑥 raised to the power seven, where 𝑎 is equal to three times 𝑒 raised to the power negative six and 𝑛 is equal to seven, applying the power rule, we have seven times three times 𝑒 raised to the power negative six times 𝑥 raised to the power 𝑛 minus one, which is seven minus one, that is, 21 times 𝑒 raised to the power negative six times 𝑥 raised to the sixth power. d𝑦 by d𝑥 is therefore 21𝑒 raised to the power negative six times 𝑥 raised to the power six.