pochodne

[xxmarcia17xx]

1) \(f(x)=lnx \sqrt{x+2}\)
2) \(g(x)= \frac{sin(x+2)}{cos(x-3)}\)
3) \(h(x)=ln(x+4)^2\)
4) \(i(x)= \sqrt{sinxlnx}\)
5) \(j(x)=(x+3x^2)^10* \sqrt[5]{x+2}\)
6) \(k(x)= \frac{sin^2 x cosx}{((e)^x)^2 +5}\)
7) \(l(x)=(x)^(3x)\)
w podpunkcie 7 jest do potęgi 3x moglabym procis o pomoc w tych zadaniach?

[domino21]

a.
\(f'(x)=\frac{1}{x} \cdot \sqrt{x+2} +\ln x \cdot\frac{1}{2\sqrt{x+2}}=\frac{\sqrt{x+2}}{x}+\frac{\ln x}{2\sqrt{x+2}}\)

[domino21]

b.
\(f'(x)=\frac{\cos(x+2)\cdot \cos(x-3)+\sin(x+2)\cdot \sin (x-3)}{\cos^2(x-3)}\)

[domino21]

c.
\(h(x)=\ln (x+4)^2=2\ln(x+4)
h'(x)=2\cdot \frac{1}{x+4}=\frac{2}{x+4}\)

[domino21]

d.
\(i(x)=\sqrt{\sin x \ln x}
i'(x)=\frac{1}{2\sqrt{\sin x \ln x}} \cdot \left( \cos x \ln x +\frac{\sin x}{ x} \right)= \frac{\cos x \ln x +\frac{\sin x}{x}}{2\sqrt{\sin x \ln x}}=\frac{x \cos x \ln x +\sin x}{2x \sqrt{\sin x \ln x}}\)

[domino21]

e.
\(j(x)=(x+3x^2)^{10} \cdot \sqrt[5]{x+2} =(x+3x^2)^{10} \cdot (x+2)^{\frac{1}{5}}
j'(x)=10(x+3x^2)^9\cdot \sqrt[5]{x+2}+\frac{1}{5}(x+2)^{-\frac{4}{5}}\cdot (x+3x^2)^{10}=10(x+3x^2)^9 \cdot \sqrt[5]{x+2} +\frac{(x+3x^2)^{10}}{5\sqrt[5]{(x+2)^4}}\)

[Galen]

zad.b
Ciąg dalszy
Wzór:
\(cos \alpha \cdot cos \beta +sin \alpha \cdot sin \beta =cos( \alpha - \beta )\)
Tu daje to możliwość:
\(\frac{cos(x+2) \cdot cos(x-3)+sin(x+2) \cdot sin(x-3)}{cos^2(x-3)}= \frac{cos(x+2-x+3)}{cos^2(x-3)}= \frac{cos5}{cos^2(x-3)}\)

[domino21]

f.
\(k(x)=\frac{\sin^2 x \cos x}{e^{2x}+5}
k'(x)=\frac{(2\sin x \cos x -\sin^3 x)\cdot (e^{2x} +5)-(\sin^2 x \cos x )\cdot 2e^{2x} }{(e^{2x}+5)^2}\)