promath The derivative of x^lnx (x to the power lnx) is equal to 2xlnx -1 lnx. Here, ln denotes the natural logarithm, that is, lnx =loge x. In this post, we will learn how to differentiate xlnx. What is the Derivative of xlnx? Answer: Explanation: To find the derivative of xlnx, we will use the logarithmic … Read more
The value of log 27 base 3 is equal to 3. Here, we learn to find the logarithm of 27 when the base is 3. The formula of log327 is as follows: log327 = 3. Find log 27 base 3 Answer: The value of log 27 with base 3 is equal to 3, that is, … Read more
The value of log 8 base 2 is equal to 3. Here, we will learn how to find logarithm of 8 with base 2. The formula of log28 is given below: log28 = 3. Value of log 8 base 2 Answer: The value of log 8 with base 2 is equal to 3, that is, … Read more
The general solution of the differential equation dy/dx=x-y is equal to y=x-1-Ce-x where C is an arbitrary constant. In this post, we will learn how to find the general solution of dy/dx =x-y. Solution of dy/dx=x-y Question: Find the genral solution of $\dfrac{dy}{dx}$ =x-y. Solution: Let x-y=v. Differentiating w.r.t x, we get that $1-\dfrac{dy}{dx}=\dfrac{dv}{dx}$ ⇒ … Read more
The general solution of the differential equation dy/dx =sec(x+y) is equal to y =tan (x+y)/2 +C where C denotes an arbitrary constant. In this post, we will learn how to find the general solution of dy/dx=sec(x+y). Solve dy/dx=sec(x+y) Question: What is the general solution of $\dfrac{dy}{dx}$ =sec(x+y)? Answer: Put x+y=v, so that $1+\dfrac{dy}{dx}=\dfrac{dv}{dx}$. ⇒ $\dfrac{dy}{dx}=\dfrac{dv}{dx}-1$ … Read more
The general solution of the differential equation dy/dx=tan(x+y) is equal to y= x -log|cos(x+y) + sin(x+y)| +C where C denotes an arbitrary constant. In this post, we will learn how to solve the differential equation dy/dx = tan(x+y). General Solution of dy/dx=tan(x+y) Question: Find the general solution of $\dfrac{dy}{dx}$=tan(x+y). Solution: $\dfrac{dy}{dx}$=tan(x+y) ⇒ $\dfrac{dy}{dx} = \dfrac{\sin(x+y)}{\cos(x+y)}$ … Read more
The tan(π-x) formula is given by tan(π-x)= -tanx. The formula of tan(pi-θ) is equal to tan(π-θ)= -tanθ. In this post, we will learn how to compute tan(pi-x) and tan(θ-pi). Note that Proof of tan(pi-x) Formula Let us use the formula: tan(a-b) = $\dfrac{\tan a -\tan b}{1+\tan a \tan b}$ …(∗) Put a=π, b=x. Thus, we … Read more
In this post, we will prove (1+tanx)(1+tany)=2 when x+y is equal to π/4 =45°. To prove this, we will use the following formula: tan(x+y) = $\dfrac{\tan x+\tan y}{1-\tan x \tan y}$. Question: If x+y=π/4, then prove that (1+tanx)(1+tany)=2. Solution: Step 1: Given that x+y=π/4. Therefore, tan(x+y) = tan(π/4) ⇒ $\dfrac{\tan x+\tan y}{1-\tan x \tan y}$ … Read more
The derivative of e^cosx (e to the power cosx) is equal to -sinx ecosx. In this post, we will find the derivative of ecosx by first principle. The derivative formula of ecosx is given below: $\dfrac{d}{dx}$(ecosx) = -sinx ecosx. Derivative of e^cosx Using First Principle By first principle, the derivative of a function f(x) using … Read more
The derivative of e^sinx is equal to cosx esinx. In this post, we will find the derivative of esinx by first principle. The esinx derivative formula is given as follows: $\dfrac{d}{dx}$(esinx) = cosx esinx. Derivative of e^sinx Using First Principle The derivative of f(x) using the first principle is given by the following limit formula: … Read more