Physics
Differentiation
Finding dy/dx from a Graph
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⚡ Quick Summary
If you have a graph of y versus x, you can find dy/dx (which is the slope of the line) at any point by drawing a tangent line at that point and calculating its slope (rise over run).
tan θ = dy/dx = (change in y) / (change in x)
The derivative, dy/dx, at a point on a curve represents the slope of the tangent line to the curve at that point. To find dy/dx graphically, draw a tangent line at the desired point on the curve. Then, calculate the slope of the tangent line using 'rise over run'.
Derivative of Area of a Square
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⚡ Quick Summary
If you increase the side of a square by a tiny amount, the area increases too. The rate at which the area changes with respect to the side length is twice the side length.
A = L², dA/dL = 2L
Consider a square with side L and area A = L². If L changes by a small amount ΔL, then A changes by ΔA. By calculating the new area (L + ΔL)² and subtracting the original area L², we find that ΔA ≈ 2LΔL. Taking the limit as ΔL approaches zero, we obtain dA/dL = 2L.
Common Derivatives
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This is a table showing derivatives of common functions.
x^n: nx^(n-1), sin x: cos x, cos x: -sin x, ln x: 1/x, e^x: e^x, tan x: sec²x, cot x: -cosec²x, cosec x: -cosec x cot x, sec x: sec x tan x
A table of derivatives of common functions is provided. This table gives the formula for calculating the derivative (dy/dx) for functions such as x^n, sin x, cos x, tan x, cot x, ln x, and e^x.
Rules for Derivatives of Composite Functions
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These are rules you can use to find derivatives of combinations of functions.
d/dx (cy) = c (dy/dx), d/dx (u + v) = du/dx + dv/dx, d/dx (uv) = u(dv/dx) + v(du/dx), d/dx (u/v) = (v(du/dx) - u(dv/dx)) / v², dy/dx = (dy/du) * (du/dx)
Several rules are given for finding the derivatives of combined functions. These include the constant multiple rule, sum rule, product rule, quotient rule, and chain rule. These rules are fundamental for differentiating more complex functions.
Example Derivative Calculation
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⚡ Quick Summary
This is an example showing how to find the derivative of ex * sin(x).
y = e^x sin x, dy/dx = e^x (cos x + sin x)
The example demonstrates how to find the derivative of y = e^x sin x using the product rule. First, identify u = e^x and v = sin x. Apply the product rule formula: d/dx (uv) = u(dv/dx) + v(du/dx). The derivatives du/dx = e^x and dv/dx = cos x. Substitute these values into the product rule formula to obtain the final result.