Differentiate the following with respect to x: 1. y = x^(1/2) + 1/x² − 3/x³ + 4x^(1/4)

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1. y = x^(1/2) − 3x^(3/4); Step 1: Power rule d/dx xⁿ = n xⁿ⁻¹. d/dx ( x^(1/2) ) = 1/2 x^(1/2−1) = 1/2 x^(−1/2) Step 2: d/dx ( −3x^(3/4) ) = −3 · 3/4 x^(3/4−1) = −9/4 x^(−1/4) Step 3: Sum the derivatives. y' = 1/2 x^(−1/2) − 9/4 x^(−1/4); y' = (1)/(2√(x)) − (9)/(4 x^(1/4)) 2. y = x − cos x − 2sin x; Step 1: d/dx x = 1, d/dx (−cos x) = sin x, d/dx (−2sin x) = −2cos x. Step 2: Sum the derivatives. y' = 1 + sin x − 2cos x; y' = 1 + sin x − 2cos x 3. y = e^(x) tan x; Step 1: Product rule d/dx (uv) = u'v + uv', where u = e^x, v = tan x. u' = e^x, v' = sec² x Step 2: Substitute. y' = e^x tan x + e^x sec² x Step 3: Factor. y' = e^x (tan x + sec² x); y' = e^(x) (tan x + sec² x) 4. y = (ln x)² − 4 ln x; Step 1: Chain rule for (ln x)²: let u = ln x, dy/du = 2u, du/dx = 1/x. d/dx (ln x)² = 2ln x · 1/x; Step 2: d/dx (−4 ln x) = −4 · 1/x. Step 3: Sum. y' = (2ln x)/(x) − 4/x = (2ln x − 4)/(x); y' = (2ln x − 4)/(x) 5. y = sin(3x + 1); Step 1: Chain rule d/dx sin u = cos u · u', u = 3x + 1, u' = 3. y' = cos(3x + 1) · 3 = 3 cos(3x + 1); y' = 3cos(3x + 1) 6. y = ln(sec x); Step 1: Chain rule d/dx ln u = 1/u u', u = sec x, u' = sec x tan x. y' = (1)/(sec x) · sec x tan x Step 2: Simplify. y' = tan x; y' = tan x 7. y = (x² + 1)⁵; Step 1: Chain rule d/dx u⁵ = 5u⁴ u', u = x² + 1, u' = 2x. y' = 5(x² + 1)⁴ · 2x = 10x (x² + 1)⁴; y' = 10x(x² + 1)⁴ 8. y = (1)/(x² + 1); Step 1: Quotient rule d/dx u/v = (u'v − uv')/(v²), u=1, v = x² + 1. u' = 0, v' = 2x Step 2: Substitute. y' = (0 · (x² + 1) − 1 · 2x)/((x² + 1)²) = (−2x)/((x² + 1)²); y' = −(2x)/((x² + 1)²) 9. y = [f(x)]⁻¹ − h(x); Step 1: Chain rule for [f(x)]⁻¹: d/dx u⁻¹ = −u⁻² u', u = f(x), u' = f'(x). d/dx [f(x)]⁻¹ = −[f(x)]⁻² f'(x) = −(f'(x))/([f(x)]²); Step 2: d/dx [−h(x)] = −h'(x). Step 3: Sum. y' = −(f'(x))/([f(x)]²) − h'(x); y' = −(f'(x))/([f(x)]²) − h'(x)

Differentiate the following with respect to x: 1. y = x^(1/2) + 1/x² - 3/x³ + 4x^(1/4)

Step-by-step solution

Power rule $\frac{d}{dx} x^{n} = n x^{n-1}$ .

Step 2

Sum the derivatives.

$\frac{d}{dx} x = 1$ , $\frac{d}{dx} (-\cos x) = \sin x$ , $\frac{d}{dx} (-2\sin x) = -2\cos x$ .

Sum the derivatives.

Product rule $\frac{d}{dx} (uv) = u'v + uv'$ , where $u = e^x$ , $v = \tan x$ .

Substitute.

Factor.

Chain rule for $(\ln x)^2$

$\frac{d}{dx} (-4 \ln x) = -4 \cdot \frac{1}{x}$ .

Sum.

Chain rule $\frac{d}{dx} \sin u = \cos u \cdot u'$ , $u = 3x + 1$ , $u' = 3$ .

Chain rule $\frac{d}{dx} \ln u = \frac{1}{u} u'$ , $u = \sec x$ , $u' = \sec x \tan x$ .

Simplify.

Chain rule $\frac{d}{dx} u^5 = 5u^4 u'$ , $u = x^2 + 1$ , $u' = 2x$ .

Quotient rule $\frac{d}{dx} \frac{u}{v} = \frac{u'v - uv'}{v^2}$ , $u=1$ , $v = x^2 + 1$ .

Substitute.

Chain rule for $[f(x)]^{-1}$

$\frac{d}{dx} [-h(x)] = -h'(x)$ .

Sum.

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Differentiate the following with respect to x: 1. y = x^(1/2) + 1/x² - 3/x³ + 4x^(1/4)

Step-by-step solution

Power rule ddxxn=nxn−1\frac{d}{dx} x^{n} = n x^{n-1}dxd​xn=nxn−1.

Step 2

Sum the derivatives.

ddxx=1\frac{d}{dx} x = 1dxd​x=1, ddx(−cos⁡x)=sin⁡x\frac{d}{dx} (-\cos x) = \sin xdxd​(−cosx)=sinx, ddx(−2sin⁡x)=−2cos⁡x\frac{d}{dx} (-2\sin x) = -2\cos xdxd​(−2sinx)=−2cosx.

Sum the derivatives.

Product rule ddx(uv)=u′v+uv′\frac{d}{dx} (uv) = u'v + uv'dxd​(uv)=u′v+uv′, where u=exu = e^xu=ex, v=tan⁡xv = \tan xv=tanx.

Substitute.

Factor.

Chain rule for (ln⁡x)2(\ln x)^2(lnx)2

ddx(−4ln⁡x)=−4⋅1x\frac{d}{dx} (-4 \ln x) = -4 \cdot \frac{1}{x}dxd​(−4lnx)=−4⋅x1​.

Sum.

Chain rule ddxsin⁡u=cos⁡u⋅u′\frac{d}{dx} \sin u = \cos u \cdot u'dxd​sinu=cosu⋅u′, u=3x+1u = 3x + 1u=3x+1, u′=3u' = 3u′=3.

Chain rule ddxln⁡u=1uu′\frac{d}{dx} \ln u = \frac{1}{u} u'dxd​lnu=u1​u′, u=sec⁡xu = \sec xu=secx, u′=sec⁡xtan⁡xu' = \sec x \tan xu′=secxtanx.

Simplify.

Chain rule ddxu5=5u4u′\frac{d}{dx} u^5 = 5u^4 u'dxd​u5=5u4u′, u=x2+1u = x^2 + 1u=x2+1, u′=2xu' = 2xu′=2x.

Quotient rule ddxuv=u′v−uv′v2\frac{d}{dx} \frac{u}{v} = \frac{u'v - uv'}{v^2}dxd​vu​=v2u′v−uv′​, u=1u=1u=1, v=x2+1v = x^2 + 1v=x2+1.

Substitute.

Chain rule for [f(x)]−1[f(x)]^{-1}[f(x)]−1

ddx[−h(x)]=−h′(x)\frac{d}{dx} [-h(x)] = -h'(x)dxd​[−h(x)]=−h′(x).

Sum.

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Power rule $\frac{d}{dx} x^{n} = n x^{n-1}$ .

$\frac{d}{dx} x = 1$ , $\frac{d}{dx} (-\cos x) = \sin x$ , $\frac{d}{dx} (-2\sin x) = -2\cos x$ .

Product rule $\frac{d}{dx} (uv) = u'v + uv'$ , where $u = e^x$ , $v = \tan x$ .

Chain rule for $(\ln x)^2$

$\frac{d}{dx} (-4 \ln x) = -4 \cdot \frac{1}{x}$ .

Chain rule $\frac{d}{dx} \sin u = \cos u \cdot u'$ , $u = 3x + 1$ , $u' = 3$ .

Chain rule $\frac{d}{dx} \ln u = \frac{1}{u} u'$ , $u = \sec x$ , $u' = \sec x \tan x$ .

Chain rule $\frac{d}{dx} u^5 = 5u^4 u'$ , $u = x^2 + 1$ , $u' = 2x$ .

Quotient rule $\frac{d}{dx} \frac{u}{v} = \frac{u'v - uv'}{v^2}$ , $u=1$ , $v = x^2 + 1$ .

Chain rule for $[f(x)]^{-1}$

$\frac{d}{dx} [-h(x)] = -h'(x)$ .