Question

X+√x×x+1 differentiate with respect to x

Answer 1

Answer

$\frac{3}{2} \sqrt{x } + 1$

Solution

Let us consider

$y = x + x\sqrt{x} + 1$

Now differentiating y with respect to x we have

$\frac{dy}{dx} = \frac{d}{dx} (x + x\sqrt{x} + 1) \\ \\ \implies \frac{dy}{dx} = \frac{dx}{dx} + \frac{d(x \sqrt{x} )}{dx} + \frac{d(1)}{dx} \\ \\ \implies \frac{dy}{dx} = 1 + \frac{d }{dx} ({x}^{1} \times {x}^{ \frac{1}{2} } ) + 0 \\ \\ \implies \frac{dy}{dx} = 1 + \frac{d}{dx} (x^{ \frac{1}{2} + 1} ) \\ \\ \implies \frac{dy}{dx} = 1 + \frac{d}{dx} ( {x}^{ \frac{3}{2} } ) \\ \\ \implies \frac{dy}{dx} = 1 + \frac{3}{2} {x}^{ (\frac{3}{2} - 1) } \\ \\ \implies \frac{dy}{dx} = \frac{3}{2} {x}^{ \frac{1}{2} } + 1 \\ \\ \implies \frac{dy}{dx} = \frac{3}{2} \sqrt{x} + 1$

Extra information

• Derivative is a rate measure i.e. it is rate of change of dependent variable with respect to independent variable .

• A renowned mathematician Leibnitz connected the concept of derivative with that of the slope of the tangent to the curve of a poiny on it .

Some formulae of Derivative :

◆ d/dx (xⁿ) = n.x^(n-1)

◆ d/dx ( sinx ) = cos x

◆d/dx (cosx) = -sinx

◆d/dx (tanx) = sec²x