Calculus Basics Notes
1. Derivative:
The derivative measures the rate of change of a function at a given point. It represents the
slope of the tangent line to the function's graph at that point. Mathematically, the derivative of a
function f(x) is denoted as f'(x) or dy/dx and is defined as:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
2. Integral:
The integral is used to compute the accumulation of a quantity over an interval. It
represents the area under the curve of a function. The definite integral of a function f(x) over an
interval [a, b] is denoted as ∫[a, b] f(x) dx and is defined as:
∫[a, b] f(x) dx = lim(n→∞) Σ[f(x_i)Δx]
where x_i are the partition points of the interval [a, b], Δx represents the width of each
subinterval, and the sum is taken over all the subintervals.
3. Chain Rule:
The chain rule is used to differentiate composite functions. It enables the differentiation
of a function within a function. Mathematically, if y = f(g(x)), the chain rule states:
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, dy/dx = dy/du * du/dx
where u = g(x) and f'(u) represents the derivative of f with respect to u.
4. Product Rule:
The product rule allows the differentiation of a product of two functions. Mathematically,
if y = f(x) * g(x), the product rule states:
d(fg)/dx = f'(x) * g(x) + f(x) * g'(x)
where f'(x) and g'(x) represent the derivatives of f(x) and g(x), respectively.
5. Quotient Rule:
The quotient rule enables the differentiation of a quotient of two functions.
Mathematically, if y = f(x) / g(x), the quotient rule states:
d(f/g)/dx = [f'(x) * g(x) - f(x) * g'(x)] / [g(x)]^2
where f'(x) and g'(x) represent the derivatives of f(x) and g(x), respectively.
6. Fundamental Theorem of Calculus:
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1. Derivative:
The derivative measures the rate of change of a function at a given point. It represents the
slope of the tangent line to the function's graph at that point. Mathematically, the derivative of a
function f(x) is denoted as f'(x) or dy/dx and is defined as:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
2. Integral:
The integral is used to compute the accumulation of a quantity over an interval. It
represents the area under the curve of a function. The definite integral of a function f(x) over an
interval [a, b] is denoted as ∫[a, b] f(x) dx and is defined as:
∫[a, b] f(x) dx = lim(n→∞) Σ[f(x_i)Δx]
where x_i are the partition points of the interval [a, b], Δx represents the width of each
subinterval, and the sum is taken over all the subintervals.
3. Chain Rule:
The chain rule is used to differentiate composite functions. It enables the differentiation
of a function within a function. Mathematically, if y = f(g(x)), the chain rule states:
This study source was downloaded by 100000888273843 from coursesidekick.com on 07-16-2024 09:07:16 GMT -05:00
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, dy/dx = dy/du * du/dx
where u = g(x) and f'(u) represents the derivative of f with respect to u.
4. Product Rule:
The product rule allows the differentiation of a product of two functions. Mathematically,
if y = f(x) * g(x), the product rule states:
d(fg)/dx = f'(x) * g(x) + f(x) * g'(x)
where f'(x) and g'(x) represent the derivatives of f(x) and g(x), respectively.
5. Quotient Rule:
The quotient rule enables the differentiation of a quotient of two functions.
Mathematically, if y = f(x) / g(x), the quotient rule states:
d(f/g)/dx = [f'(x) * g(x) - f(x) * g'(x)] / [g(x)]^2
where f'(x) and g'(x) represent the derivatives of f(x) and g(x), respectively.
6. Fundamental Theorem of Calculus:
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