promath The integral of e^(-x) from 0 to infinity is equal to 1. Here we will learn how to integrate e-x (e to the power -x) from 0 to ∞. Answer: The integral of e-x from 0 to infinity is equal to 1. That is, $\int_0^\infty e^{-x} dx =1$. Explanation: The integration of e-x from 0 … Read more
The derivative of ln4x is equal to 1/x. The derivative of ln4x is denoted by d/dx(ln4x), and its formula is given by $\dfrac{d}{dx}(\ln 4x)=\dfrac{1}{x}$. Find the Derivative of ln4x Answer: The derivative of ln(4x) is 1/x. Explanation: As ln4x = ln4 + lnx, the derivative of ln 4x is equal to $\dfrac{d}{dx}$ (ln 4x) = … Read more
The derivative of ln2x is 1/x, where ln denotes the natural logarithm. Here, the ln2x derivative is computed using the first principle and the chain rule of derivatives. Note that the derivative of ln2x is denoted by d/dx(ln2x), and its formula is given as follows: $\dfrac{d}{dx}(\ln 2x)=\dfrac{1}{x}.$ That is, the differentiation of ln2x is given … Read more
The derivative of lnx^2 is equal to 2/x. The natural logarithm of x2 is denoted by ln(x2), and its derivative formula is given by $\dfrac{d}{dx}(\ln x^2)=\dfrac{2}{x}.$ That is, the differentiation of ln(x2) is 2/x. Derivative of ln(x2) by Chain Rule Answer: The derivative of ln(x2) is 2/x. Explanation: To find the derivative of ln(x2) by … Read more
The derivative of ln(x+1) is equal to 1/(x+1). Ln(x+1) denotes the natural logarithm of x+1, that is, ln(x+1) = loge(x+1). Here we will find the derivative of ln(x+1) using the following methods: The derivative of ln(x+1) is denoted by d/dx {ln(x+1)} and its formula is given by $\dfrac{d}{dx} \Big(\ln (x+1) \Big)=\dfrac{1}{x+1}.$ Derivative of ln(x+1) by … Read more
The derivative of lnx is equal to 1/x. Ln(x) denotes the natural logarithm of x, that is, lnx= logex. Here we will find the derivative of ln(x) using the limit definition and chain rule of differentiation. Note that lnx= logex. The derivative of lnx is denoted by d/dx(lnx), and its formula is given as follows: … Read more
The derivative of ln u is equal to 1/u du/dx, and this is the derivative of ln(u) with respect to x. Here we differentiate the natural logarithm of u with respect to x. Recall that, ln u = logeu. Derivative of ln(u) If u is a function of x, then the derivative of ln u … Read more
The derivative of ln3x is equal to 1/x, where ln denotes the natural logarithm. Here we will find the derivative of ln(3x). The ln3x differentiation formula is given below: $\dfrac{d}{dx}(\ln 3x)=\dfrac{1}{x}$. ln(3x) Derivative Question: Find the derivative of ln3x. Solution: By the logarithm rule for products, ln 3x = ln3 + lnx. So, $\dfrac{d}{dx}(\ln 3x)$ … Read more
The derivative of tan root x is equal to (sec2√x)/2√x. By the chain rule of differentiation, the derivative formula of tan(√x) with respect to x is given as follows: $\dfrac{d}{dx}(\tan \sqrt{x})=\dfrac{\sec^2 \sqrt{x}}{2\sqrt{x}}$. Derivative of tan(√x) by Chain Rule Answer: The derivative of tan√x is (sec2√x)/2√x. Explanation: Let us put z = √x. Differentiating both sides … Read more
The derivative of cos root x is equal to (-sin√x)/2√x. Here we will find the derivative of cos√x by the chain rule of derivatives. The differentiation of cos(√x) with respect to x is given by $\dfrac{d}{dx}(\cos \sqrt{x})=-\dfrac{\sin \sqrt{x}}{2\sqrt{x}}$. Let us now find the derivative of cos root x using the chain rule of differentiation. Derivative … Read more