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differential equation full notes { mhtcet }
Welcome to the World of Differential Maths
Imagine you're on a road trip, and you're trying to figure out how far you've traveled and how fast
you're going
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But what if
you wanted to know your average speed over the entire trip? Or what if you wanted to know exactly
where you were 30 minutes ago? That's where differential maths comes in
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It's like having a superpower that allows you to analyze and understand how things change and
move
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A limit is a value that a function approaches as
the input (or independent variable) gets arbitrarily close to a certain point
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As you get
closer and closer, your distance from the wall gets smaller and smaller
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Calculating Limits: Step by Step
Let's calculate the limit of the function f(x) = 1/x as x approaches 2
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1
f(2
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1
Step 3: Evaluate the function at x = 2
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01) = 1/2
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Derivatives: The Rate of Change
Now that we have limits, we can talk about derivatives
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Think of it like this:
Imagine you're driving a car, and you're going uphill
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The steeper the hill, the faster you're going
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Using the power rule, which states that if f(x) = x^n, then f'(x) = nx^(n-1), we get:
f'(x) = 2x
So, the derivative of x^2 is 2x
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(Just Kidding!)
We've just scratched the surface of differential maths
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Stay tuned
Explicit and Implicit Functions: Unveiling the Mysteries
Imagine you're on a hike, and you stumble upon a hidden trail
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The map represents an explicit function, where the relationship
between the variables is clearly defined
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Explicit Functions
An explicit function is a mathematical relationship where the output variable is directly expressed in
terms of the input variable(s)
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Example 1:
Let's say we have an explicit function: y = 2x + 3
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For every value of x, we can calculate the corresponding value of y using the function
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It's like trying to find the hidden trail without
a map; you need to use subtle hints and clues to navigate
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Here, we can't directly express y in terms of x
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Step-by-Step Calculation:
1
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y^2 = 4 - x^2
3
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Code Sample:

differential equation full notes { mhtcet }
In Python, we can use the sympy library to solve implicit functions:
import sympy as sp

x, y = sp
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solve(eq, y)

print(y_sol) # Output: [-sqrt(4 - x**2), sqrt(4 - x**2)]
Hand-Drawn Plot:
Imagine a circle with radius 2, centered at the origin
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For every value of x, there are two corresponding values of y that satisfy the equation
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While explicit functions provide a clear and direct path, implicit functions require more
work and creativity to solve
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One way to describe your motion is by using parametric functions, which define your position and
velocity as functions of time
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Suppose you're driving in a straight line, and your position (s) is
given by the equation s(t) = 2t^2 + 3t + 1, where t is time in hours
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Calculating Derivatives
To calculate the derivative of a parametric function, we use the chain rule
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Define the parametric function: x(t) = 2t^2 + 3t + 1 and y(t) = t^2 - 2t + 1
2
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Find the derivative of y with respect to x: dy/dx = (dy/dt) / (dx/dt) = (2t - 2) / (4t + 3)
Example Walkthrough
Suppose we want to find the derivative of the parametric curve x(t) = 2t^2 + 3t + 1 and y(t) = t^2 - 2t
+ 1 at t = 2
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Evaluate the derivatives: dx/dt = 4(2) + 3 = 11 and dy/dt = 2(2) - 2 = 2
2
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At t = 2, the curve
has a slope of 2/11, which represents the instantaneous rate of change of y with respect to x
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symbols('t')
x = 2*t**2 + 3*t + 1
y = t**2 - 2*t + 1

dxdt = sp
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diff(y, t)

dydx = dydt / dxdt

print(dydx)
This code will output the derivative of y with respect to x as a function of t
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•

The derivative of a parametric function represents the instantaneous rate of change of the
curve
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That's where logarithmic differentiation comes in, a powerful
technique that helps you find the derivative of complicated functions with ease
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This may seem like a trivial step, but it
can simplify even the most complicated functions
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But with logarithmic differentiation, we can simplify the process
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Take the logarithm of both sides:
$$\ln(f(x)) = \ln\left(\frac{(2x+1)^3}{(x-1)^2}\right)$$
2
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Differentiate both sides with respect to x:
$$\frac{1}{f(x)} \cdot f'(x) = 3 \cdot \frac{2}{2x+1} - 2 \cdot \frac{1}{x-1}$$
4
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symbols('x')
f = ((2*x+1)**3)/((x-1)**2)

# Take the logarithm of both sides
log_f = sp
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diff(log_f, x)

# Solve for f'(x)
f_diff = f * log_f_diff

differential equation full notes { mhtcet }

print(f_diff)
Hand-Drawn Plot
Let's visualize the function and its derivative to get a better understanding of what's happening:
In this plot, we can see that the original function has a few sharp turns, which makes it difficult to
differentiate
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Example 2: A More Complicated Function
Suppose we want to find the derivative of the function:
$$f(x) = \frac{(x^2+2x+1)^2}{(x^2-4x+3)^3}$$
Using logarithmic differentiation, we can simplify the process and obtain the derivative in just a few
steps:
1
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Apply the logarithm properties to simplify the expression:
$$\ln(f(x)) = 2\ln(x^2+2x+1) - 3\ln(x^2-4x+3)$$
3
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Solve for $f'(x)$:
$$f'(x) = f(x) \cdot \left( \frac{4x+4}{x^2+2x+1} - \frac{6x-12}{x^2-4x+3} \right)$$
As you can see, logarithmic differentiation can simplify even the most complicated functions, making
it a powerful tool in your calculus toolkit

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