There are two things to understand with this question: 'differentiate' and 'with respect to x'we differentiate when we need to find how quickly something changes, which in maths usually means the gradient of a curve'with respect to x' means we want to see how y is affected by a change in x. In an exam, all you need to know is that you focus on the side of the equation with x terms
y = 2x⁴ + 11x² - 5
--> for this question, we need to look at each term individually. A term is a number, a variable (such as x or y), or a combination of the two.There are 3 terms we need to focus on: 2x⁴ , 11x² , and - 5
an exponent is the small number after the x, and means the number of times you multiply x with itselfa coefficient is the number before the x
How to differentiate: multiply the exponent of x with its coefficient (think as multiplying the numbers either side of x). This becomes the new coefficient. Then, take away 1 from the exponent. This is the new term.
For 2x⁴ :the exponent is 4the coefficient is 2to differentiate, start by multiplying these numbers: 8. This is the new coefficientthen subtract 1 from the current exponent, 4, to get 3. This is the new exponentThe differentiated term is 8x³
For 11x², the same steps are used:multiply 2 with 11 to get 22 as the new coefficientsubtract 1 from the exponent, 2, to get 1. In maths, we don't write an exponent if it is 1, because we don't need to (e.g. 7¹ is the same thing as 7)so the differentiated term is 22x
-5 is different. As a shortcut, know that a number becomes 0 when differentiated. This is because when you differentiate, you only care about what changes when x changes. -5 doesn't change when x changes- it stays as -5, which means it doesn't affect the rate, so we can get rid of it
Now combine all the differentiated terms to get the final answer, replace y with dy/dx to show you have differentiated:
dy/dx = 8x³ + 22x

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