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Calculus requires the use of derivatives. They assist us in determining the rate of change, maxima, and minima for the functions. Derivatives are defined by employing limits, which is known as the initial form of the derivative.
The chain rule of differentiation may be used to derive derivatives of composite functions. When one function is expressed in terms of another, the result is a composite function. This means that in a composite function, one function may be swapped for another, and is commonly represented as (c o d)(x) = c(d(x)). To get the derivatives of the composite functions, we first differentiate the first function with respect to the second function, and then the second function w.r.t the variable, i.e., (c o d)'(x) = c'(d(x)). d'(x).
The formula for Derivatives of Composite Functions
The derivative of a composite function and second order derivatives h(x) = f(g(x)) may be calculated by multiplying the derivative of f(x) with respect to g(x) by the derivative of g(x) with respect to the variable x. The equation for the derivatives of composite functions is expressed numerically as:
Composite Function Derivatives in a Single Variable
The simple chain rule formula is used to calculate derivatives of composite functions in one variable.
Composite Functions Derivatives in Two Variables
The derivative of a function with numerous variables is computed one variable at a time. Such derivatives are referred to as partial derivatives. Using the chain rule technique of differentiation for one variable, we may compute the partial derivatives of composite functions z = h(x, y). When calculating the partial derivative of a function with respect to one variable, we treat all other variables as constants.
Chain Rule and Composite Functions
Assume you have a function f(x) = (x + 1)2 that we wish to compute the derivative for. These functions are known as composite functions since they are made up of more than one function. They are usually of the form g(x) = h(f(x)), although they may also be expressed as g = hof (x). In this situation, the supplied function f(x) = (x + 1)2 is made up of two functions,
f(x)=g(h(x))
Where,
g(x)=x2 and h(x)=x+1
Second Order Derivatives
The derivative of a function is known as a second-order derivative. The slope of the tangent at that position or the instantaneous rate of change of a function at that point are both represented by the first-order derivative at that location. Second-Order Derivative offers us an understanding of the form of a particular function’s graph. The second derivative of the function f(x) is often abbreviated as f” (x). If y = f, it is sometimes expressed as y2 or y” (x).
Second-Order Derivatives of a Parametric Function
We utilise the chain rule twice to determine the second derivative of the function in parametric form. To determine the second derivative, first find the derivative of the first derivative w.r.t t and then divide by the derivative of x w.r.t t. If x = x(t) and y = y(t), then its Second Order Parametric Form is:
First Derivative: dy/dx=(dy/dt)/(dx/dt)
Second Derivative: d2y/dx2=d/dx(dy/dx) = d/dt(dy/dx)/(dx/dt)
Second-Order Derivatives Graphical Representation
The slope of the function at a point is represented graphically by the first derivative, and the slope varies over the independent variable in the graph by the second derivative.
Function Concavity
Let f(x) be a differentiable function in an appropriate interval. The graph of f(x) may therefore be classified as follows:
Concave Up: A portion of a curve is concave upwards if the y-value increases at an increasing pace from left to right.
Concave Down: Concave down is the inverse of concave upwards, in which the y-value drops from left to right.
Inflexion Point
The point of inflexion is a point on the function’s graph when the graph shifts from upward to downward concave or downward to upward concave. At this moment, the sign of the second-order derivative is likewise shifted from positive towards negative or from negative towards positive. At such a point, the function’s second-order derivative is generally regarded as zero.
A function with a second derivate determines the inflexion point values at the local maximum and lowest. These may be recognised using the following criteria:
If f”(a) is equal to 0, the function f(a) has a local maximum at a.
If f”(a) is greater than 0, the function f(a) has a local minimum at a.
If f”(a) is equal to 0, then nothing can be concluded about point a.
Conclusion
In layman’s terms, the derivative of a composite function and second-order derivatives are the product of the derivatives of the outer function and the inner function with respect to the variable.
A second-order derivative is a derivative of a function’s derivative. It is calculated using the first-order derivative. So we obtain the derivative of a function first, and then the derivative of the first derivative. The concavity and inflexion points may be determined using a second-order derivative.
