Use differentiation rules to determine the derivatives of each of the following, leaving the answer with…

Question

Use differentiation rules to determine the derivatives of each of the following, leaving the answer with positive exponent(s) and in surd form where applicable:

ScanSolve AI Tutor · Accepted Answer

Here are the derivatives for each function:

3.2.1
Given the function:

y = 4e^{\frac{x}{2}} + \frac{1}{2}\ln x - \frac{3}{\sqrt{x}} + \sin 30^\circ

Rewrite terms with exponents for easier differentiation.
$y = 4e^{\frac{x}{2}} + \frac{1}{2}\ln x - 3x^{-\frac{1}{2}} + \sin 30^\circ$
Differentiate each term using the appropriate rules ( $\frac{d}{dx}(e^{ax}) = ae^{ax}$ , $\frac{d}{dx}(\ln x) = \frac{1}{x}$ , power rule, and derivative of a constant is 0).
$\frac{dy}{dx} = 4 \left(\frac{1}{2}e^{\frac{x}{2}}\right) + \frac{1}{2}\left(\frac{1}{x}\right) - 3\left(-\frac{1}{2}x^{-\frac{3}{2}}\right) + 0$
Simplify the expression and write with positive exponents and surd form.
$\frac{dy}{dx} = 2e^{\frac{x}{2}} + \frac{1}{2x} + \frac{3}{2}x^{-\frac{3}{2}}$ $\frac{dy}{dx} = 2e^{\frac{x}{2}} + \frac{1}{2x} + \frac{3}{2x\sqrt{x}}$
The final answer is:
$\boxed{\frac{dy}{dx} = 2e^{\frac{x}{2}} + \frac{1}{2x} + \frac{3}{2x\sqrt{x}}}$
3.2.2
Given the function:
$y = \frac{6x^4 - 3x^2}{x^2}$
Simplify the function by dividing each term in the numerator by $x^2$ .
$y = \frac{6x^4}{x^2} - \frac{3x^2}{x^2}$ $y = 6x^2 - 3$
Differentiate the simplified function using the power rule.
$\frac{dy}{dx} = 6(2x) - 0$ $\frac{dy}{dx} = 12x$
The final answer is:
$\boxed{\frac{dy}{dx} = 12x}$
3.2.3
Given the function:
$y = 3x^3 \sin(2x)$
Use the product rule $\frac{d}{dx}(uv) = u'v + uv'$ .
Let $u = 3x^3$ and $v = \sin(2x)$ .
Find the derivatives of $u$ and $v$ .
$u' = \frac{d}{dx}(3x^3) = 9x^2$ $v' = \frac{d}{dx}(\sin(2x)) = \cos(2x) \cdot \frac{d}{dx}(2x) = 2\cos(2x)$
Apply the product rule.
$\frac{dy}{dx} = (9x^2)(\sin(2x)) + (3x^3)(2\cos(2x))$
Simplify the expression.
$\frac{dy}{dx} = 9x^2\sin(2x) + 6x^3\cos(2x)$
The final answer is:
$\boxed{\frac{dy}{dx} = 9x^2\sin(2x) + 6x^3\cos(2x)}$
3.2.4
Given the function:
$y = \frac{3}{(2-x^2)^4}$
Rewrite the function using a negative exponent.
$y = 3(2-x^2)^{-4}$
Use the chain rule $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$ .
Let $f(u) = 3u^{-4}$ and $u = 2-x^2$ .
$f'(u) = 3(-4)u^{-5} = -12u^{-5}$ $g'(x) = \frac{d}{dx}(2-x^2) = -2x$
Apply the chain rule.
$\frac{dy}{dx} = (-12(2-x^2)^{-5})(-2x)$
Simplify the expression and write with positive exponents.
$\frac{dy}{dx} = 24x(2-x^2)^{-5}$ $\frac{dy}{dx} = \frac{24x}{(2-x^2)^5}$
The final answer is:
$\boxed{\frac{dy}{dx} = \frac{24x}{(2-x^2)^5}}$
3.2.5
Given the function:
$y = \ln(\cos 3x)$
Use the chain rule for logarithmic functions $\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$ .
Here $f(x) = \cos 3x$ .
Find the derivative of $f(x)$ .
$f'(x) = \frac{d}{dx}(\cos 3x) = -\sin(3x) \cdot \frac{d}{dx}(3x) = -3\sin(3x)$
Apply the chain rule.
$\frac{dy}{dx} = \frac{-3\sin(3x)}{\cos(3x)}$
Simplify using the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ .
$\frac{dy}{dx} = -3\tan(3x)$
The final answer is:
$\boxed{\frac{dy}{dx} = -3\tan(3x)}$
3.2.6
Given the function:
$y = -3e^{5x} + \frac{2}{5x^{-2}} - \frac{\sqrt{x^5}}{2} + \frac{\pi x}{2}$
Rewrite terms with exponents for easier differentiation.
$y = -3e^{5x} + \frac{2}{5}x^2 - \frac{1}{2}x^{\frac{5}{2}} + \frac{\pi}{2}x$
Differentiate each term using the appropriate rules ( $\frac{d}{dx}(e^{ax}) = ae^{ax}$ and power rule).
$\frac{dy}{dx} = -3(5e^{5x}) + \frac{2}{5}(2x) - \frac{1}{2}\left(\frac{5}{2}x^{\frac{3}{2}}\right) + \frac{\pi}{2}(1)$
Simplify the expression and write with positive exponents and surd form.
$\frac{dy}{dx} = -15e^{5x} + \frac{4}{5}x - \frac{5}{4}x^{\frac{3}{2}} + \frac{\pi}{2}$ $\frac{dy}{dx} = -15e^{5x} + \frac{4}{5}x - \frac{5}{4}x\sqrt{x} + \frac{\pi}{2}$
The final answer is:
$\boxed{\frac{dy}{dx} = -15e^{5x} + \frac{4}{5}x - \frac{5}{4}x\sqrt{x} + \frac{\pi}{2}}$

Use differentiation rules to determine the derivatives of each of the following, leaving the answer with positive exponent(s) and in surd form where applicable:

Step-by-step solution

Rewrite terms with exponents for easier differentiation.

Differentiate each term using the appropriate rules ( $\frac{d}{dx}(e^{ax}) = ae^{ax}$ , $\frac{d}{dx}(\ln x) = \frac{1}{x}$ , power rule, and derivative of a constant is 0).

Simplify the expression and write with positive exponents and surd form.

Simplify the function by dividing each term in the numerator by $x^2$ .

Differentiate the simplified function using the power rule.

Use the product rule $\frac{d}{dx}(uv) = u'v + uv'$ .

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

Apply the product rule.

Simplify the expression.

Rewrite the function using a negative exponent.

Use the chain rule $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$ .

Apply the chain rule.

Simplify the expression and write with positive exponents.

Use the chain rule for logarithmic functions $\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$ .

Find the derivative of $f(x)$ .

Apply the chain rule.

Simplify using the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ .

Rewrite terms with exponents for easier differentiation.

Differentiate each term using the appropriate rules ( $\frac{d}{dx}(e^{ax}) = ae^{ax}$ and power rule).

Simplify the expression and write with positive exponents and surd form.

Still stuck on this one?

More Mathematics Questions

Use differentiation rules to determine the derivatives of each of the following, leaving the answer with positive exponent(s) and in surd form where applicable:

Step-by-step solution

Rewrite terms with exponents for easier differentiation.

Differentiate each term using the appropriate rules (ddx(eax)=aeax\frac{d}{dx}(e^{ax}) = ae^{ax}dxd​(eax)=aeax, ddx(ln⁡x)=1x\frac{d}{dx}(\ln x) = \frac{1}{x}dxd​(lnx)=x1​, power rule, and derivative of a constant is 0).

Simplify the expression and write with positive exponents and surd form.

Simplify the function by dividing each term in the numerator by x2x^2x2.

Differentiate the simplified function using the power rule.

Use the product rule ddx(uv)=u′v+uv′\frac{d}{dx}(uv) = u'v + uv'dxd​(uv)=u′v+uv′.

Find the derivatives of uuu and vvv.

Apply the product rule.

Simplify the expression.

Rewrite the function using a negative exponent.

Use the chain rule ddx(f(g(x)))=f′(g(x))⋅g′(x)\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)dxd​(f(g(x)))=f′(g(x))⋅g′(x).

Apply the chain rule.

Simplify the expression and write with positive exponents.

Use the chain rule for logarithmic functions ddx(ln⁡(f(x)))=f′(x)f(x)\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}dxd​(ln(f(x)))=f(x)f′(x)​.

Find the derivative of f(x)f(x)f(x).

Apply the chain rule.

Simplify using the identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ​.

Rewrite terms with exponents for easier differentiation.

Differentiate each term using the appropriate rules (ddx(eax)=aeax\frac{d}{dx}(e^{ax}) = ae^{ax}dxd​(eax)=aeax and power rule).

Simplify the expression and write with positive exponents and surd form.

Still stuck on this one?

More Mathematics Questions

Differentiate each term using the appropriate rules ( $\frac{d}{dx}(e^{ax}) = ae^{ax}$ , $\frac{d}{dx}(\ln x) = \frac{1}{x}$ , power rule, and derivative of a constant is 0).

Simplify the function by dividing each term in the numerator by $x^2$ .

Use the product rule $\frac{d}{dx}(uv) = u'v + uv'$ .

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

Use the chain rule $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$ .

Use the chain rule for logarithmic functions $\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$ .

Find the derivative of $f(x)$ .

Simplify using the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ .

Differentiate each term using the appropriate rules ( $\frac{d}{dx}(e^{ax}) = ae^{ax}$ and power rule).