One of the basics that used in the calculus.
- Power Differentiation
- Find the Gradient of the Curve
- Trigonometric Differentiation
What is differentiation? Differentiation is to find the rate of change, for example when in the graph, there is x plane and y plane. Differentiation for that graph is rate of change of x over y, also known as dy/dx.
The usage of differentiation also happened in finding the velocity (speed with a direction) over time, that is acceleration (the unit would be m/s²). Displacement divided by time is velocity, while velocity divided by time is acceleration.
Power Differentiation
This power differentiation means that a number with a variable (usually x), and the power would be on the variable, then the number with powered variable is differentiated. The example is:
y=5x³
dy/dx= 15x²
Why the 5x³ becomes 15x² after differentiation? The explanation is in the formula below:
If y= axⁿ , then dy/dx= anxⁿ⁻¹
For the formulas above, a is any number, x is a variable, and n is the power number. Let's do another example:
y= 6x⁴+ 5x²
dy/dx= (6)(4)x⁴⁻¹ + (5)(2)x²⁻¹ = 24x³+10x
Another example is below:
The picture above shows the differentiation process. The first step is to eliminate the x^(4/3) at the denominator. Division for the normal number is as subtraction for the powers. Thus, the power 10 is minus by power 4/3 to get power 26/3, while power 3 is minus by power 4/3 to get power 5/3. Thus, y is 12x power of 26/3, plus 5x power of 5/3. When in differentiation, the number is multiplied by the power number, and power numbers are decreased by 1. Thus, the differentiation of y is 104x power of 23/3, plus 25/3 multiplied by x power of 2/3.
Find the Gradient of the Curve
What is the gradient of the curve? The gradient is a straight line on a point of the curve, and gradient is the tangent of that point. Gradient is also known as slope. The example is the picture below (The picture is from BestMaths Website about gradient):
The gradient of the line at the curve, given at the point P is 1/2 or 0.5. Let's see how to find the gradient on a curve, below:
y=2x² <- The formula of the curve.
dy/dx= 4x <-gradient
Let the point of gradient= (2,8). The gradient would be 4x= (4)(2)= 8.
Insert the gradient to line formula:
y=mx+c
y=8x+c
Use the point of gradient to find y intersection (c)
8=8(2)+c
c=8-16=-8
Thus, the gradient line is:
y=8x-8
The picture of the gradient touch the curve at (2,8) is shown below:
Based on the picture above, the blue line is gradient line, while red line is curve line.
Trigonometric Differentiation
y= sin x -> dy/dx= cos x
y= cos x -> dy/dx= -sin x
y= tan x -> dy/dx= sec² x
What happened when the angle component (x in sin x or cos x or tan x) has power on it? Differentiate it on the outside. Example:
y= cos x³ -> dy/dx= (3x²)(-sin x³)= -3x²sin x³
Another example on the trigonometric differentiation:
y= 2sin x+ 3cos x⁵- 5tan x
dy/dx= 2cos x+ (5x⁴)(-3sin x⁵)- 5sec² x
= 2cos x- 15x⁴sin x⁵- 5sec² x #
The blog of basic differentiation is finished here, if any comment or idea please comment below. Thank you!
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