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
The derivative of sinx/(1+cosx) [sinx divided by 1+cosx] is equal to 1/(1+cosx), and it is denoted by d/dx (sinx/1+cosx). That is, $\dfrac{d}{dx} \big(\dfrac{\sin x}{1+\cos x} \big)=\dfrac{1}{1+\cos x}$. Here we will learn how to differentiate sinx divided by 1+cosx. Differentiate sinx/(1+cosx) Question: If y= sinx/1+cosx, then find dy/dx. Answer: If y=sinx/(1+cosx), then dy/dx = 1/(1+cosx). Explanation: … Read more
The derivative of cosx/x (cosx divided by x) is equal to -(xsinx +cosx)/x2. Here we will learn to differentiate cosx/x by the quotient rule as well as implicit differentiation method. Let us now differentiate cosx/x. Derivative of cosx/x by Quotient Rule By the quotient rule, the derivative of $\dfrac{f}{g}$ is given by $\dfrac{d}{dx}\big( \dfrac{f}{g}\big)=\dfrac{gf’ -fg’}{g^2}$ … Read more
The derivative of x^1/x (x to the power 1/x) is denoted by d/dx(x1/x) and its value is equal to x1/x[1/x2 – 1/x2 ⋅ logex]. The derivative formula of x1/x is given below: d/dx(x1/x) = x1/x -2 [1 – logex]. Let us now learn how to differentiate x1/x. Derivative of x to the power 1/x Question: … Read more
The derivative of x^tanx (x to the power tanx) is denoted by d/dx(xtanx) and its value is equal to xtanx[tanx/x + sec2x logex]. The derivative formula of xtanx is given by d/dx(xtanx) = xtanx[tanx/x + sec2x logex]. Let us now learn how to differentiate xtanx. Differentiate xtanx Question: Prove that d/dx(xtanx) = xtanx[tanx/x + sec2x … Read more
The derivative of x^π (x to the power pi) is equal to πxπ-1. In this post, we will learn how to differentiate x to the power pi with respect to x. The derivative of xπ is denoted by d/dx (xπ) or (xπ)’. Its formula is given by $\dfrac{d}{dx}(x^\pi)=\pi x^{\pi -1}$. Derivative of x to the … Read more
The derivative of x^e (x to the power e) is equal to exe-1. Here we will learn to differentiate x^e with respect to x. The derivative of xe is denoted by d/dx (xe) or (xe)’. Its formula is given by $\dfrac{d}{dx}(x^e)=ex^{e-1}$. Derivative of x to the power e Question: Find the derivative of xe, that … Read more
The derivative of x^cosx (x to the power cosx) is equal to xcosx[cosx/x – sinx logex]. Here we learn how to differentiate x^cosx. The derivative of xcosx is denoted by d/dx(xcosx) and its formula is given below: d/dx(xcosx) = xcosx[cosx/x – sinx logex]. Let us now give a proof of xcosx derivative. Differentiate xcosx Question: … Read more