Differentiate y = 2x^4 + 11x^2 - 5 with respect to x
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
