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Given $y = (4x - 1)^2$, find \( \frac{dy}{dx} \). - Scottish Highers Maths - Question 3 - 2017

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Given-$y-=-(4x---1)^2$,-find-\(-\frac{dy}{dx}-\).-Scottish Highers Maths-Question 3-2017.png

Given $y = (4x - 1)^2$, find \( \frac{dy}{dx} \).

Worked Solution & Example Answer:Given $y = (4x - 1)^2$, find \( \frac{dy}{dx} \). - Scottish Highers Maths - Question 3 - 2017

Step 1

Start to differentiate

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Answer

To differentiate the function y=(4x−1)2y = (4x - 1)^2, we apply the chain rule. The chain rule states that if you have a composition of functions, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

Step 2

Complete differentiation

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Answer

First, differentiate the outer function: ( \frac{d}{dx}[(u)^2] = 2u \cdot \frac{du}{dx} ), where ( u = 4x - 1 ). Now, find ( \frac{du}{dx} ): ( \frac{du}{dx} = 4 ). So, the derivative becomes:

dydx=2(4x−1)⋅4=8(4x−1)\frac{dy}{dx} = 2(4x - 1) \cdot 4 = 8(4x - 1)

Finally, simplifying gives:

dydx=32x−8.\frac{dy}{dx} = 32x - 8.

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