Differentiation Cheat Sheet - Ch.12 - First Principle, Tangents and Normals

Oct 13, 2026

Differentiation is presented as a method to compute the gradient (slope) of a function at different points and to analyze how one variable changes with respect to another. The notes introduce derivative notation, relate gradients to tangents, and derive the derivative from first principles using limits. Worked examples show differentiating power forms and quadratics, finding gradients at specific points, and determining equations of tangents and normals. Increasing and decreasing criteria via the derivative sign are also included.

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Differentiation Cheat Sheet

Derivative notation and gradient of curves

Derivative from first principles

Differentiating power functions and examples

Differentiating quadratics and gradient at a point

Gradient, tangents and normal

Increasing and decreasing functions

Differentiation Cheat Sheet Differentiation is a process that helps us to calculate gradient or slope of a function at different points. It also help us to identify change in one variable with respect to another variable. You will learn real life application of differentiation in this topic Derivative Notation: !!"\# or 􀀢$ (􀀤) represents derivative of 􀀦 = 􀀢(􀀤) with respect to x Gradient of curves: Unlike straight lines, the gradient of a curve changes constantly. The gradient of a curve at any given point is the same as calculating gradient of the tangent at that given point. You can find exact gradient of a curve at any given point using derivatives. Finding derivative: In this section you will learn how to find derivative of a function (i.e. exact gradient of a curve or function at a given point) Consider the following figure for a curve 􀀦 = 􀀢 (􀀤) As point 􀁓 moves closer to point 􀁆 , the gradient of chord 􀁆􀁓 gets closer to the gradient of tangent to the curve at 􀁆 . The coordinate of 􀁓 is (􀀣$, 􀀡 (􀀣$)) and 􀁕 is (􀀣$ + ℎ, 􀀡 (􀀣$ + ℎ)) . So, the gradient of AB is-(($0"")*-(($) . As ℎ gets smaller, gradient of 􀁓􀁕 gets closer to gradient of curve at 􀁓 . Hence, The gradient function or the derivative of a curve 􀀷 = 􀀢 (􀀤) is given by 􀁎! (􀁏) = 􀁔􀁕􀁖 􀀱→􀀲 􀀳(􀀴0􀀱)*􀀳(􀀴)􀀱 lim "→$ means the limit as ℎ tends to 0. This rule is called differentiating from the first principle Example 1: 􀀢 (􀀤) = 􀀤 % a. Show that 􀀢 $ (􀀤) = lim (2􀀤 + ℎ) b. Hence deduce that 􀀢 $ (􀀤) = 2􀀤 &→( a. Use the definition of derivative 􀀡 ! (􀀣) = lim 􀀡 (􀀣 + ℎ) − 􀀡(􀀣) "→$ ℎ So 􀀡(􀀣) = 􀀣 % implies 􀀡 (􀀣 + ℎ) = (􀀣 + ℎ)%, Substitute 􀀡 (􀀣 + ℎ) and 􀀡(􀀣) into the above definition of derivative (􀀣 + ℎ)% − 􀀣 %􀀡 ! (􀀣) = lim ~~ ~~ "→$ ℎ 􀀡 ! (􀀣) = lim "→$ 􀀣 % + 2􀀣ℎ + ℎ% − 􀀣 % ℎ Expand the bracket (􀀢 + ℎ)% = (􀀢 + ℎ)(􀀢 += 􀀢 % + 2􀀢ℎ + Factorise the numerator ↓ 􀀡 ! (􀀣) = lim 2􀀣ℎ + ℎ% ⟹ 􀀡 ! (􀀣) "→$ ℎ 􀀡 ! (􀀣) = lim(2􀀣 + ℎ) "→$ Differentiating 􀀡􀀡 = lim "→$ ℎ(2􀀣 + ℎ) ℎ ℎ) ℎ% b. 􀀢 ! (􀀤) = 2􀀤 From part a, we know that 􀀢 ! (􀀤) = lim (2􀀤 + ℎ) "→$ Apply the limits As h→0, 2􀀤 + ℎ → 2􀀤 h→0 means as h tends to 0 or h approaches to 0, 2􀀤 + ℎ approaches 2􀀤 Hence, 􀀢 ! (􀀤) = lim (2􀀤 + ℎ) = 2􀀤 "→$ In this chapter you will only learn how to differentiate functions of the form 􀀣&, where 􀀯 is any number. Use the following results to differentiate functions. 􀀰 = 􀀡 (􀀣) = 􀀣& then ''() = 􀀡 ! (􀀣) = 􀀯􀀣&*+ where 􀀯 is any real number 􀀰 = 􀀡 (􀀣) = 􀀱􀀣& then ''() = 􀀡 ! (􀀣) = 􀀱􀀯􀀣&*+ where 􀀯 is any real number and 􀀱 is a constant Here is an example for you to understand how to apply the results Example 2: Find the derivative, 􀀡! (􀀣) when 􀀡(􀀣) equals: a. 􀀣 , b. 10􀀣*+ a. 􀀡 (􀀣) = 􀀣 , You are supposed to find derivative of 􀀡(􀀣) = 􀀣 ,, use the result for 􀀣& So 􀀡! (􀀣) = 6􀀣,*+ = 6􀀣- b. 􀀡 (􀀣) = 10􀀣*+ Use the result for 􀀱􀀣& 􀀡 ! (􀀣) = 10 × (−1) × 􀀣*+*+ = −10 􀀣 *% 􀀡 ! (􀀣) = −10 × )+! = − $!~~ ~~ As 􀀣 *% = )+! Differentiating quadratics: A quadratic function or a curve is given by 􀀷 = 􀀸􀀤% + 􀀹􀀤 + 􀀺, where 􀀸, 􀀹 and 􀀺 are constants. The derivative of 􀀷 is &&'( = 2􀀸􀀤 + 􀀹 Note: Differentiate each term one at a time Derivative of only a constant term is always 0. So if 􀀷 = 2 then &&'( = 0 Example 3: Find the gradient of the curve with equation 􀀷 = 2􀀤% − 􀀤 − 1 at the point (2, 5) As explained in the gradient of curves section, finding the gradient of a curve at a point is same as calculating derivative of the curve at that point. So first, you will have to calculate the gradient of 􀀷 = 2􀀤% − 􀀤 − 1 &&'( = 2 × 2 × 􀀤(%*+) − 1 × 􀀤(+*+) − 0 Since 1 is a constant, derivative of a constant will be 0 &&'( = 4􀀤 − 􀀤 $ = 4􀀤 − 1 As 􀀤 $ = 1 Now to find the derivative of 􀀷 at point(2, 5), you need to substitute 􀀤 = 2 in the derivative function 􀁂􀁂􀀷􀀤 = 􀀢 ! (2) = 4 (2) − 1 = 8 − 1 = 7 Hence, the gradient of the curve with equation 􀀷 = 2􀀤% − 􀀤 − 1 at the point (2, 5) is 7. Gradient, tangents and normal

FAQs

How is differentiation used to find the gradient of a curve?

The gradient of a curve at a point equals the gradient of the tangent at that point. Differentiation provides that exact gradient by evaluating the derivative at the chosen coordinate.

What does “differentiating from the first principle” mean?

It means using the definition of the derivative based on limits of the difference quotient as h tends to 0. The notes show substituting values into this limit to obtain the derivative.

How do you find the equations of a tangent and a normal to a curve at a point?

Compute the derivative to get the tangent gradient at the point, then use the tangent line equation with point and slope. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal, and it uses the same point with that slope.

Oct 13, 2026

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