Calculus

Derivative of xe^-x by First Principle & Product Rule

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The derivative of xe-x is equal to (1-x)e-x. In this post, we will find the derivative of xe^-x by the product rule and the first principle of derivatives. Derivative of xe-x by First Principle The derivative of $f(x)=xe^{-x}$by the first principle is given by $(xe^{-x})’$ = limh→0 $\dfrac{(x+h)e^{-(x+h)}-xe^{-x}}{h}$ = limh→0 $\dfrac{xe^{-(x+h)}+he^{-(x+h)}-xe^{-x}}{h}$ = limh→0 $\big(\dfrac{xe^{-(x+h)}-xe^{-x}}{h}$ $+\dfrac{he^{-(x+h)}}{h} … Read more

Derivative of 1/x^2 by First Principle

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The derivative of 1 by x square is equal to -2/x3. In this blog post, we will learn how to find the differentiation of 1/x2. What is the Derivative of 1/x2? Let us now find the derivative of 1/x^2 by the power rule of derivatives. The power rule says that $\dfrac{d}{dx}(x^n)=nx^{n-1}$ Step 1: First,we write … Read more

[Solved] What is the Derivative of 1/(1+x^2)?

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The derivative of 1 by 1+x^2 is equal to -2x/(1+x^2)^2. In this post, we find the derivative of 1/(1+x^2) using the chain rule and substitution method. Derivative of 1/(1+x^2) Let us find the derivative of 1/(1+x^2) by the substitution method and the product rule of derivatives. Suppose that $u=\dfrac{1}{1+x^2}$ …(I) We will need to find … Read more

Derivative of e^x^2 by First Principle and Chain Rule

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The function e to the power x2 is written as $e^{x^2}$ and its derivative is $2xe^{x^2}$. In this post, we will find the derivative of e to the power x square by the first principle and chain rule of derivatives. Recall the first principle of derivatives: The derivative of a function f(x) by first principle … Read more

Derivative of sin^2x by First Principle | sin^2x Derivative

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The derivative of sin^2x by first principle is equal to 2sinx cosx=sin2x. Sin square x derivative is denoted by d/dx (sin2x) and its formula is given as follows: $\dfrac{d}{dx}$(sin2x) = 2sinx cosx = sin2x. In this post, we will find the derivative of sine square x. Derivative of sin square x by First Principle The … Read more

Product Rule of Limits: Proof, Examples [Epsilon-Delta Method]

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The product rule of limits says that the limit of the product of two functions is the same as the product of the limits of the individual functions. In this post, we will prove the product law of limits by the epsilon-delta method. Question: What is an epsilon-delta proof of the product law of limits? … Read more

Sum Rule of Limits: Proof and Examples [ε-δ Method]

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The sum rule of limits says that the limit of the sum of two functions is the same as the sum of the limits of the individual functions. In this post, we will prove the sum/addition rule of limits by the epsilon-delta method.   Question: What is an epsilon-delta proof of the sum law of … Read more