Question

Differentiate the following functions with respect to x (a) x^(3) + 5x^(2) - 2 (b) x sin x (c) (2x + 3)^(6) (d) (x)/ (sinx) (e) e^((5x+2))

Answer 1

(a) 3x^2 + 10x

(b)xcosx+sinx

(C)6(2x+3)^5(2)

(D)-xsinx^-2cosx + sinx^-1

(E) 5e^(5x+2)

Answer 2

$\text{(a)}$

$y=x^3+5x^2-2$

$\implies\bf\frac{dy}{dx}=3x^2+10x$

$\text{(b)}$

$y=x\;sinx$

$\text{Using product rule of differentiation}$

$\boxed{\bf\frac{d(uv)}{dx}=u\;\frac{dv}{dx}+v\;\frac{du}{dx}}$

$\frac{dy}{dx}=x(cosx)+sinx(1)$

$\implies\bf\frac{dy}{dx}=x\,cosx+sinx$

$\text{(c)}$

$y=(2x+3)^6$

$\frac{dy}{dx}=6(2x+3)^5(2)$

$\implies\bf\frac{dy}{dx}=12(2x+3)^5$

$\text{(d)}$

$y=\frac{x}{sinx}$

$\text{Using quotient rule of differentiation}$

$\boxed{\bf\frac{d(u/v)}{dx}=\frac{v\,\frac{du}{dx}-u\,\frac{dv}{dx}}{v^2}}$

$\frac{dy}{dx}=\frac{sinx(1)-x(cosx)}{sin^2x}$

$\implies\bf\frac{dy}{dx}=\frac{sinx-x\,cosx}{sin^2x}$

$\text{(e)}$

$y=e^{5x+2}$

$\frac{dy}{dx}=e^{5x+2}(5)$

$\implies\bf\frac{dy}{dx}=5e^{5x+2}$