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Examples on Chain Rule (Differentiation Rules)

Examples on Chain Rule Example 10.8 Find F′(x) if F (x) = √[x2 +1] Solution Take u = g(x) = x2 + 1 and f (u) =√u ∴ F (x) = ( fog)(x) = f (g(x)) Example 10.9 Differentiate…

Euler’s Homogeneous Function Theorem

Let  be a homogeneous function of order  so that (1) Then define  and . Then (2) (3) (4) Let , then (5) This can be generalized to…

Homogeneous Functions

To be Homogeneous a function must pass this test: f(zx,zy) = znf(x,y) In other words Homogeneous is when we can take a function:f(x,y)…

Higher-Order Partial Derivatives

Fred has mastered differentiating when a function has only one variable. Somehow, he missed the lessons on partial differentiation. He’s…

Geometry of partial derivatives

Partial Derivatives

Now that we have the brief discussion on limits out of the way we can proceed into taking derivatives of…

Limits and Continuity

The concept of the Limits and Continuity is one of the most crucial things to understand in order to prepare for calculus.…

Indeterminate Forms and l’Hospital’s Rule

In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule, uses derivatives to calculate limits.…

Taylor’s Theorem

One of the most important uses of infinite series is the potential for using an initial portion of the series…

Maclaurin’s Theorem

This simple article shows how the Maclaurin series works, and how to write out the expansions. Although modern calculators are…

Leibnitz Theorem

Leibnitz Theorem is basically the Leibnitz rule defined for derivative of the antiderivative. As per the rule, the derivative on nth…

Trigonometric Function Differentiation

The six trigonometric functions also have differentiation formulas that can be used in application problems of the derivative. The rules…