Differentiation or derivative
Quiz 1
Differentiation Rules and Concepts
Gradient of a Curve
The gradient of a curve is represented as:
dy/dx
This represents the rate of change of the curve \( y = f(x) \).
The Four Rules of Differentiation
1. Power Rule
If \( y = x^n \), then:
d/dx (x^n) = n * x^(n-1)
2. Scalar Multiple Rule
If \( y = kf(x) \), where \( k \) is a constant, then:
d/dx [kf(x)] = k * d/dx [f(x)]
3. Addition/Subtraction Rule
If \( y = f(x) ± g(x) \), then:
d/dx [f(x) ± g(x)] = d/dx [f(x)] ± d/dx [g(x)]
4. Chain Rule
If \( y = f(u) \) and \( u = g(x) \), then:
dy/dx = (dy/du) * (du/dx)
Tangents and Normals
If the value of \( dy/dx \) at the point \( (x_1, y_1) \) is \( m \), then:
- The equation of the tangent at that point is given by:
- The equation of the normal at that point is given by:
y - y₁ = m(x - x₁)
y - y₁ = -1/m (x - x₁)
Second Derivatives
The second derivative is defined as:
d/dx (dy/dx) = d²y/dx²
Differentiation Advanced Concepts
Product Rule
If \( y = u(x) \cdot v(x) \), then the derivative is:
d/dx [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)
Quotient Rule
If \( y = \frac{u(x)}{v(x)} \), then the derivative is:
d/dx [u(x)/v(x)] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
Implicit Differentiation
When a function is not explicitly solved for \( y \), differentiation is performed with respect to \( x \), treating \( y \) as a function of \( x \). For example:
- Given \( x^2 + y^2 = r^2 \), differentiate both sides with respect to \( x \):
- Solve for \( dy/dx \):
2x + 2y(dy/dx) = 0
dy/dx = -x/y
Higher-Order Derivatives
Higher-order derivatives are obtained by differentiating the first derivative. For example:
- The second derivative is:
- The third derivative is:
d²y/dx² = d/dx (dy/dx)
d³y/dx³ = d/dx (d²y/dx²)
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