promath The limit of (1-cosx)/x^2 as x approaches 0 is equal to 1/2. That is, limx→0 (1-cosx)/x2 =1/2. This limit can be computed using the formula limx→0 $\dfrac{\sin x}{x}$ = 1 …(∗) Prove that limx→0 (1-cosx)/x2 =1/2 We have limx→0 $\dfrac{1-\cos x}{x^2}$ = limx→0 $\dfrac{2 \sin^2 \frac{x}{2}}{x^2}$ using the formula 1-cos2x=2sin2x. = limx→0 $\dfrac{2 \sin^2 \frac{x}{2}}{(\frac{x}{2})^2 … Read more
The derivative of sin(xy) is equal to (y+x dy/dx) cos(xy), and this is the derivative of sin(xy) with respect to x. The derivative of sin(xy) formula is given below: $\dfrac{d}{dx}(\sin xy)=(y+x\dfrac{dy}{dx})\cos xy$. Differentiate sin(xy) with respect to x Answer: The derivative of sin(xy) with respect to x is equal to (y+x dy/dx) cos(xy). Explanation: Let … Read more
If y=cos(x+y), then dy/dx= -sin(x+y)/[1+sin(x+y)]. Here, we learn how to differentiate y=cos(x+y) with respect to x. Let us find the derivative of y=cos(x+y). Find dy/dx if y=cos(x+y) Question: If y=cos(x+y), then $\dfrac{dy}{dx}$. Solution: We are given that y = cos(x+y). Step 1: Differentiating both sides of the above equation with respect x, we get that … Read more
If y=sin(x+y), then dy/dx= cos(x+y)/[1-cos(x+y)]. Here, we learn how to differentiate y=sin(x+y) with respect to x. Let us find the derivative of y=sin(x+y). y=sin(x+y), Find dy/dx Question: If y=sin(x+y), then $\dfrac{dy}{dx}$. Solution: Given, y = sin(x+y). To find dy/dx, we will differentiate both sides of the equation y=sin(x+y) with respect x. Using the chain rule, … Read more
The integral of cos root x dx is denoted by ∫cos(√x)dx, and it is equal to ∫cos(√x)dx = 2[√xsin(√x)+ cos(√x)]+C where C is an integration constant. Here we will learn how to integrate cos root x. The integral formula of cos root x is given below. $\int \cos \sqrt{x} dx$ $= 2[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]+C$ … Read more
The integral of sin root x dx is denoted by ∫sin(√x)dx, and it is given by ∫sin(√x)dx = 2[-√xcos(√x)+ sin(√x)]+C where C denotes an integral constant. Note that $\int \sin \sqrt{x} dx$ $= 2[-\sqrt{x} \cos \sqrt{x}+\sin \sqrt{x}]+C$ Lets learn how to integrate sin(sqrt x) dx. Integration of sin root x Question: Find the integral ∫sin(√x)dx. … Read more
The limit of (1+1/x)^x as x approaches infinity is equal to e. Here we will discuss Lim x→∞ (1+(1/x))^x formula with proof. Note that Limx → ∞ (1+$\frac{1}{x}$)x = e. Limx→∞ (1+(1/x))x Formula with Proof The formula of limx→∞ (1+(1/x))x is given by limx → ∞ (1+(1/x))x =e. Explanation: Let y = limx→∞ (1+$\frac{1}{x}$)x. So … Read more
The limit of cosx/x as x approaches 0 does NOT exist. That is, limx→0 (cos x)/x is undefined. $\lim\limits_{x \to 0} \dfrac{\cos x}{x}$ = NOT exist. Let us now show that the limit of (cos x)/x when x tends to 0 does not exist. Limx→0 (cos x)/x Question: What is the limit of cosx/x when … Read more
The limit of cosx/x as x approaches infinity is equal to 0. The formula of limx→∞ (cos x)/x is given below: $\lim\limits_{x \to \infty} \dfrac{\cos x}{x}=0$. Here we find the limit of (cos x)/x when x tends to infinity. Limx→∞ (cos x)/x = 0 Proof Question: What is the limit of cosx/x when x tends … Read more
The integral of 1 is equal to x+C where C is a constant. The integration of 1 is denoted by ∫1 dx, and its formula is given as follows: ∫1 dx = x+C where C is an arbitrary integration constant. Here we will learn how to integrate 1. Integration of 1 Question: What is the … Read more