Derivadas

Encuenta la derivada.

Ejercicio 1

1. \(f(x)=3x^5 - 4x^3 - 4\)
2. \(f(x)=x + \ln x\)
3. \(f(x)=2x^2 - e^2\)
4. \(f(x)=x e^x\)
5. \(f(x)=x\sqrt{x}\)
6. \(f(x)=\frac{x+2}{x-2}\)
7. \(f(x)=\frac{\ln x}{e^x}\)
8. \(f(x)=e^{2x}+\frac{2}{x}\)
9. \(f(x)=(2x+1)^3\)
10. \(f(x)=2^{2x}\)
11. \(f(x)=e^{2-3x}\)
12. \(f(x)=(2x-1)\,e^{\,2x-1}\)
13. \(f(x)=\sqrt{2x-1}\)
14. \(f(x)=\ln(x^2 - 2)\)
15. \(f(x)=x^2(2x-1)^3\)
16. \(f(x)=\frac{e^x}{x+1}\)
17. \(f(x)=\frac{x^2-1}{\sqrt{x-1}}\)
18. \(f(x)=\ln\!\left(\frac{x^2}{x+1}\right)\)

Soluciones

Ejercicio 2

1. \(f(x)=\dfrac{e^{5x}}{x^3 - 1}\)
2. \(f(x)=4x\ln(3x+1)\)
3. \(f(x)=(x^2 - 1)(x^3 + 2x)\)
4. \(f(x)=\dfrac{e^{2x+1}}{(x-1)^2}\)
5. \(f(x)=\ln\!\left(\dfrac{x}{x+1}\right)\)
6. \(f(x)=\dfrac{3x-1}{x}\,(5x - x^2)^2\)
7. \(f(x)=(x^2 - 1)\ln x\)
8. \(f(x)=2^{5x}\)
9. \(f(x)=(x^3 - 6x)(x^2 + 1)^3\)
10. \(f(x)=(x+1)e^{2x+1}\)
11. \(f(x)=\dfrac{\ln x}{x^2}\)
12. \(f(x)=(1 - x^3)\cos x\)

Soluciones

Ejercicio 3

1. \(f(x)=2x^3 - 3x^2 + 2\)
2. \(f(x)=\frac{2}{3}x^3 - \frac{1}{2}x^2 + 2\)
3. \(f(x)=x^{2}-\ln x + e^x\)
4. \(f(x)=x^2 e^{2x}\)
5. \(f(x)=e^x\sqrt{x}\)
6. \(f(x)=x^2 \ln x\)
7. \(f(x)=\frac{2}{x^2 - 1}\)
8. \(f(x)=\frac{e^x}{x}\)
9. \(f(x)=e^{2x}+\frac{2}{x^2}\)
10. \(f(x)=(x^2 - 2)^2\)
11. \(f(x)=3^{x^2+1}\)
12. \(f(x)=2x^2 e^{x^2}\)
13. \(f(x)=\sqrt{x^2 + 1}\)
14. \(f(x)=\ln\sqrt{2x - 1}\)
15. \(f(x)=2x^2( x^2 - 2)^3\)
16. \(f(x)=\frac{2x - 1}{\sqrt{x}}\)
17. \(f(x)=\frac{\ln x^2}{e^{2x}}\)
18. \(f(x)=\ln\frac{1}{e^x}\)

Soluciones

Ejercicio 4

1. \(f(x)=\frac{1-3x}{x} + (5x-2)^3\)
2. \(f(x)=(x^2+2)\ln(x^2+2)\)
3. \(f(x)=3^{5x} + e^x\)
4. \(f(x)=\frac{3}{(2x-5)^2} + \ln(1-x)\)
5. \(f(x)=\frac{e^x}{x^3+1}\)
6. \(f(x)=(3x+1)^3 \ln(x^2+1)\)
7. \(f(x)=\frac{e^x}{7x^5 - 4}\)
8. \(f(x)=(x^3+1)e^{7x}\)
9. \(f(x)=3^x \ln(x)\)
10. \(f(x)=(x^2+1)(x^5-6x)^6\)
11. \(f(x)=\frac{(x+1)^2}{x^2 - 2}\)
12. \(f(x)=4x^3 - 5x + \frac{1}{e^x}\)

Soluciones

Ejercicio 5. Calcula aplicando la regla del Hôpital.

1. \(\displaystyle \lim_{x\to 1} \frac{x^3-2x+1}{4x^2-2x-2}\)
2. \(\displaystyle \lim_{x\to 0} \frac{\sin 2x}{x}\)
3. \(\displaystyle \lim_{x\to 0} \frac{\ln(x+1)}{x^2+2x}\)
4. \(\displaystyle \lim_{x\to 0} \frac{\ln(2x+1)}{\sin x}\)
5. \(\displaystyle \lim_{x\to +\infty} \frac{e^x}{\ln x}\)
6. \(\displaystyle \lim_{x\to 0} \frac{\tan x - x}{x - \sin x}\)
7. \(\displaystyle \lim_{x\to 1} \left(\frac{x}{x-1} - \frac{1}{\ln x}\right)\)
8. \(\displaystyle \lim_{x\to 0} \frac{1-\sqrt{1-x^2}}{x^2}\)

Soluciones