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Given that π¦ equals negative seven [π‘] cubed plus eight and π§ equals negative seven [π‘] squared plus three, find the rate of change of π¦ with respect to π§.
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When faced with a question about the rate of change of something, we should be thinking about derivatives.
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Here we want to find the rate of change of π¦ with respect to π§.
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So weβre going to work out dπ¦ by dπ§.
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Thatβs the first derivative of π¦ with respect to π§.
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We then recall that, given two parametric equations β π₯ is equal to π of π‘ and π¦ is equal to π of π‘ β we find dπ¦ by dπ₯ by multiplying dπ¦ by dπ‘ by one over dπ₯ by dπ‘.
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Or equivalently, by dividing dπ¦ by dπ‘ by dπ₯ by dπ‘.
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In this example, our two functions are π¦ and π§.
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So we say that dπ¦ by dπ§ equals dπ¦ by dπ‘ divided by dπ§ by dπ‘.
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And we see that weβre going to need to begin by differentiating each function with respect to π‘.
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Weβll begin by differentiating π¦ with respect to π‘.
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Remember, to differentiate a polynomial term, we multiply the term by the exponent and then reduce that exponent by one.
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So the first derivative of negative 70 cubed is three times negative 70 squared.
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And actually, the first derivative of eight is zero.
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Of course, we donβt really need to include that plus zero.
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So we find that dπ¦ by dπ‘ is equal to negative 21π‘ squared.
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Weβll now repeat this for dπ§ by dπ‘.
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This time, the first derivative is two times negative seven π‘, which is negative 14π‘.
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dπ¦ by dπ§ is what we get when we divide dπ¦ by dπ‘ by dπ§ by dπ‘.
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So thatβs negative 21π‘ squared divided by negative 14π‘.
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Of course, a negative divided by a negative is a positive.
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And we can divide both the numerator and the denominator by π‘.
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Our final step is to simplify by dividing by 21 and 14 by seven.
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So we find the rate of change of π¦ with respect to π§ to be three π‘ over two.