Differentiate the following functions: 1. y = x3 + x 2. y = x2 x 3. y = (x2 + x)/(x)

Question

ScanSolve AI Tutor · Accepted Answer

Here are the differentiations for each function:

$y = x^3 + \sin x$

Differentiate each term using the power rule and the derivative of $\sin x$ .

$$
\frac{dy}{dx} = \frac{d}{dx}(x^3) + \frac{d}{dx}(\sin x)
$$
$$
\frac{dy}{dx} = 3x^{3-1} + \cos x
$$
$$
\frac{dy}{dx} = 3x^2 + \cos x
$$
The derivative is $\boxed{3x^2 + \cos x}$.

2. $y = x^2 \cos x$

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2$ and $v = \cos x$ .

Find the derivatives of $u$ and $v$ .

$u' = \frac{d}{dx}(x^2) = 2x$  
$v' = \frac{d}{dx}(\cos x) = -\sin x$

Substitute $u, v, u', v'$ into the product rule formula.

$$
\frac{dy}{dx} = (2x)(\cos x) + (x^2)(-\sin x)
$$
$$
\frac{dy}{dx} = 2x \cos x - x^2 \sin x
$$
The derivative is $\boxed{2x \cos x - x^2 \sin x}$.

3. $y = \frac{x^2 + \sin x}{x}$

Apply the quotient rule, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$ , where $u = x^2 + \sin x$ and $v = x$ .

Find the derivatives of $u$ and $v$ .

$u' = \frac{d}{dx}(x^2 + \sin x) = 2x + \cos x$  
$v' = \frac{d}{dx}(x) = 1$

Substitute $u, v, u', v'$ into the quotient rule formula.
$$ \frac{dy}{dx} = \frac{(2x + \cos x)(x) - (x^2 + \sin x)(1)}{x^2} $$

Simplify the expression.

$$
\frac{dy}{dx} = \frac{2x^2 + x \cos x - x^2 - \sin x}{x^2}
$$
$$
\frac{dy}{dx} = \frac{x^2 + x \cos x - \sin x}{x^2}
$$
The derivative is $\boxed{\frac{x^2 + x \cos x - \sin x}{x^2}}$.

4. $y = (x^2 + 1)(\sin x)$

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2 + 1$ and $v = \sin x$ .

Find the derivatives of $u$ and $v$ .

$u' = \frac{d}{dx}(x^2 + 1) = 2x$  
$v' = \frac{d}{dx}(\sin x) = \cos x$

Substitute $u, v, u', v'$ into the product rule formula.

$$
\frac{dy}{dx} = (2x)(\sin x) + (x^2 + 1)(\cos x)
$$
$$
\frac{dy}{dx} = 2x \sin x + (x^2 + 1) \cos x
$$
The derivative is $\boxed{2x \sin x + (x^2 + 1) \cos x}$.

5. $y = x^3 \tan x$

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^3$ and $v = \tan x$ .

Find the derivatives of $u$ and $v$ .

$u' = \frac{d}{dx}(x^3) = 3x^2$  
$v' = \frac{d}{dx}(\tan x) = \sec^2 x$

Substitute $u, v, u', v'$ into the product rule formula.

$$
\frac{dy}{dx} = (3x^2)(\tan x) + (x^3)(\sec^2 x)
$$
$$
\frac{dy}{dx} = 3x^2 \tan x + x^3 \sec^2 x
$$
The derivative is $\boxed{3x^2 \tan x + x^3 \sec^2 x}$.

6. $y = \frac{\cos x}{x^2}$

Apply the quotient rule, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$ , where $u = \cos x$ and $v = x^2$ .

Find the derivatives of $u$ and $v$ .

$u' = \frac{d}{dx}(\cos x) = -\sin x$  
$v' = \frac{d}{dx}(x^2) = 2x$

Substitute $u, v, u', v'$ into the quotient rule formula.
$$ \frac{dy}{dx} = \frac{(-\sin x)(x^2) - (\cos x)(2x)}{(x^2)^2} $$

Simplify the expression.

$$
\frac{dy}{dx} = \frac{-x^2 \sin x - 2x \cos x}{x^4}
$$
$$
\frac{dy}{dx} = \frac{-x(x \sin x + 2 \cos x)}{x^4}
$$
$$
\frac{dy}{dx} = \frac{-(x \sin x + 2 \cos x)}{x^3}
$$
The derivative is $\boxed{\frac{-(x \sin x + 2 \cos x)}{x^3}}$.

7. $y = x \sin x + x^2$

Differentiate each term.
For $x \sin x$ , apply the product rule.
For $u = x$ and $v = \sin x$ :
$u' = 1$
$v' = \cos x$ $\frac{d}{dx}(x \sin x) = (1)(\sin x) + (x)(\cos x) = \sin x + x \cos x$
Differentiate $x^2$ .
$$ \frac{d}{dx}(x^2) = 2x $$

Combine the derivatives.

$$
\frac{dy}{dx} = (\sin x + x \cos x) + 2x
$$
$$
\frac{dy}{dx} = \sin x + x \cos x + 2x
$$
The derivative is $\boxed{\sin x + x \cos x + 2x}$.

8. $y = (x^2 + x) \cos x$

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2 + x$ and $v = \cos x$ .

Find the derivatives of $u$ and $v$ .

$u' = \frac{d}{dx}(x^2 + x) = 2x + 1$  
$v' = \frac{d}{dx}(\cos x) = -\sin x$

Substitute $u, v, u', v'$ into the product rule formula.

$$
\frac{dy}{dx} = (2x + 1)(\cos x) + (x^2 + x)(-\sin x)
$$
$$
\frac{dy}{dx} = (2x + 1) \cos x - (x^2 + x) \sin x
$$
The derivative is $\boxed{(2x + 1) \cos x - (x^2 + x) \sin x}$.

9. $y = \frac{x^3}{\sin x}$

Apply the quotient rule,

Differentiate the following functions: 1. y = x3 + x 2. y = x2 x 3. y = (x2 + x)/(x)

Step-by-step solution

Differentiate each term using the power rule and the derivative of $\sin x$ .

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2$ and $v = \cos x$ .

Find the derivatives of $u$ and $v$ .

Substitute $u, v, u', v'$ into the product rule formula.

Apply the quotient rule, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$ , where $u = x^2 + \sin x$ and $v = x$ .

Find the derivatives of $u$ and $v$ .

Substitute $u, v, u', v'$ into the quotient rule formula.

Simplify the expression.

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2 + 1$ and $v = \sin x$ .

Find the derivatives of $u$ and $v$ .

Substitute $u, v, u', v'$ into the product rule formula.

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^3$ and $v = \tan x$ .

Find the derivatives of $u$ and $v$ .

Substitute $u, v, u', v'$ into the product rule formula.

Apply the quotient rule, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$ , where $u = \cos x$ and $v = x^2$ .

Find the derivatives of $u$ and $v$ .

Substitute $u, v, u', v'$ into the quotient rule formula.

Simplify the expression.

Differentiate each term.

Differentiate $x^2$ .

Combine the derivatives.

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2 + x$ and $v = \cos x$ .

Find the derivatives of $u$ and $v$ .

Substitute $u, v, u', v'$ into the product rule formula.

Apply the quotient rule,

Still stuck on this one?

More Mathematics Questions

Differentiate the following functions: 1. y = x3 + x 2. y = x2 x 3. y = (x2 + x)/(x)

Step-by-step solution

Differentiate each term using the power rule and the derivative of sin⁡x\sin xsinx.

Apply the product rule, ddx(uv)=u′v+uv′\frac{d}{dx}(uv) = u'v + uv'dxd​(uv)=u′v+uv′, where u=x2u = x^2u=x2 and v=cos⁡xv = \cos xv=cosx.

Find the derivatives of uuu and vvv.

Substitute $u, v, u', v'$ into the product rule formula.

Apply the quotient rule, ddx(uv)=u′v−uv′v2\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}dxd​(vu​)=v2u′v−uv′​, where u=x2+sin⁡xu = x^2 + \sin xu=x2+sinx and v=xv = xv=x.

Find the derivatives of uuu and vvv.

Substitute $u, v, u', v'$ into the quotient rule formula.

Simplify the expression.

Apply the product rule, ddx(uv)=u′v+uv′\frac{d}{dx}(uv) = u'v + uv'dxd​(uv)=u′v+uv′, where u=x2+1u = x^2 + 1u=x2+1 and v=sin⁡xv = \sin xv=sinx.

Find the derivatives of uuu and vvv.

Substitute $u, v, u', v'$ into the product rule formula.

Apply the product rule, ddx(uv)=u′v+uv′\frac{d}{dx}(uv) = u'v + uv'dxd​(uv)=u′v+uv′, where u=x3u = x^3u=x3 and v=tan⁡xv = \tan xv=tanx.

Find the derivatives of uuu and vvv.

Substitute $u, v, u', v'$ into the product rule formula.

Apply the quotient rule, ddx(uv)=u′v−uv′v2\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}dxd​(vu​)=v2u′v−uv′​, where u=cos⁡xu = \cos xu=cosx and v=x2v = x^2v=x2.

Find the derivatives of uuu and vvv.

Substitute $u, v, u', v'$ into the quotient rule formula.

Simplify the expression.

Differentiate each term.

Differentiate x2x^2x2.

Combine the derivatives.

Apply the product rule, ddx(uv)=u′v+uv′\frac{d}{dx}(uv) = u'v + uv'dxd​(uv)=u′v+uv′, where u=x2+xu = x^2 + xu=x2+x and v=cos⁡xv = \cos xv=cosx.

Find the derivatives of uuu and vvv.

Substitute $u, v, u', v'$ into the product rule formula.

Apply the quotient rule,

Still stuck on this one?

More Mathematics Questions

Differentiate each term using the power rule and the derivative of $\sin x$ .

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2$ and $v = \cos x$ .

Find the derivatives of $u$ and $v$ .

Apply the quotient rule, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$ , where $u = x^2 + \sin x$ and $v = x$ .

Find the derivatives of $u$ and $v$ .

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2 + 1$ and $v = \sin x$ .

Find the derivatives of $u$ and $v$ .

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^3$ and $v = \tan x$ .

Find the derivatives of $u$ and $v$ .

Apply the quotient rule, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$ , where $u = \cos x$ and $v = x^2$ .

Find the derivatives of $u$ and $v$ .

Differentiate $x^2$ .

Apply the product rule, $\frac{d}{dx}(uv) = u'v + uv'$ , where $u = x^2 + x$ and $v = \cos x$ .

Find the derivatives of $u$ and $v$ .