WEBVTT
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Given that π¦ is equal to three times π₯ raised to the power seven times π raised to the negative six, determine dπ¦ by dπ₯.
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Weβre given that π¦ is equal to three times π₯ raised to the seventh power times π to the negative six, where π is Eulerβs number, which to five decimal places is 2.71828.
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And weβre asked to find the derivative of π¦ with respect to π₯.
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The first thing to note is that π¦ is a function of π₯ is not as complicated as it looks.
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In fact, since both three and π to the negative six are constants, if we rearrange a little so that our constants are together, we have an expression of the form π¦ is equal to π times π₯ raised to the πth power.
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In our case, π is equal to three times π raised to the power negative six and π is equal to seven.
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Weβre asked to determine dπ¦ by dπ₯.
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And for a function of this type, we can use the power rule for differentiation.
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This says that if π¦ is equal to π times π₯ raised to the power π, dπ¦ by dπ₯ is equal to π times π times π₯ raised to the power π minus one.
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That is, we multiply by the exponent π and subtract one from the exponent to get π minus one as our new exponent.
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And remembering that, in our case, π¦ is equal to three times π raised to the power negative six times π₯ raised to the power seven, where π is equal to three times π raised to the power negative six and π is equal to seven, applying the power rule, we have seven times three times π raised to the power negative six times π₯ raised to the power π minus one, which is seven minus one, that is, 21 times π raised to the power negative six times π₯ raised to the sixth power. dπ¦ by dπ₯ is therefore 21π raised to the power negative six times π₯ raised to the power six.