Munoz Donnelly
04/12/2024 · Junior High School
allar las derivadas mediante limites de las siguientes funcione: \( \begin{array}{ll}\text { 1. } f(x)=3 x^{2}+5 & \text { 2. } f(x)=2 x^{3}-15 x \\ \text { 3. } f(x)=-4 x^{2}+19 x & \text { 4. } f(x)=12 x+16 x \\ \text { 5. } f(x)=7 x^{3}-5 & \text { 6. } f(x)=-6 x^{3}+9 x^{2} \\ \text { 7. } f(x)=x^{3}+6 x^{2}+11 x & \end{array} \)
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Tutor-Verified Answer
Step-by-step Solution
Para encontrar las derivadas de las funciones dadas mediante el límite, utilizamos la definición de la derivada:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]
A continuación, calcularemos la derivada de cada función.
### 1. \( f(x) = 3x^2 + 5 \)
\[
f(x+h) = 3(x+h)^2 + 5 = 3(x^2 + 2xh + h^2) + 5 = 3x^2 + 6xh + 3h^2 + 5
\]
\[
f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 + 5) - (3x^2 + 5) = 6xh + 3h^2
\]
\[
f'(x) = \lim_{h \to 0} \frac{6xh + 3h^2}{h} = \lim_{h \to 0} (6x + 3h) = 6x
\]
### 2. \( f(x) = 2x^3 - 15x \)
\[
f(x+h) = 2(x+h)^3 - 15(x+h) = 2(x^3 + 3x^2h + 3xh^2 + h^3) - 15x - 15h
\]
\[
= 2x^3 + 6x^2h + 6xh^2 + 2h^3 - 15x - 15h
\]
\[
f(x+h) - f(x) = (2x^3 + 6x^2h + 6xh^2 + 2h^3 - 15x - 15h) - (2x^3 - 15x) = 6x^2h + 6xh^2 + 2h^3 - 15h
\]
\[
f'(x) = \lim_{h \to 0} \frac{6x^2h + 6xh^2 + 2h^3 - 15h}{h} = \lim_{h \to 0} (6x^2 + 6xh + 2h - 15) = 6x^2 - 15
\]
### 3. \( f(x) = -4x^2 + 19x \)
\[
f(x+h) = -4(x+h)^2 + 19(x+h) = -4(x^2 + 2xh + h^2) + 19x + 19h
\]
\[
= -4x^2 - 8xh - 4h^2 + 19x + 19h
\]
\[
f(x+h) - f(x) = (-4x^2 - 8xh - 4h^2 + 19x + 19h) - (-4x^2 + 19x) = -8xh - 4h^2 + 19h
\]
\[
f'(x) = \lim_{h \to 0} \frac{-8xh - 4h^2 + 19h}{h} = \lim_{h \to 0} (-8x - 4h + 19) = -8x + 19
\]
### 4. \( f(x) = 12x + 16x = 28x \)
\[
f(x+h) = 28(x+h) = 28x + 28h
\]
\[
f(x+h) - f(x) = (28x + 28h) - 28x = 28h
\]
\[
f'(x) = \lim_{h \to 0} \frac{28h}{h} = \lim_{h \to 0} 28 = 28
\]
### 5. \( f(x) = 7x^3 - 5 \)
\[
f(x+h) = 7(x+h)^3 - 5 = 7(x^3 + 3x^2h + 3xh^2 + h^3) - 5
\]
\[
= 7x^3 + 21x^2h + 21xh^2 + 7h^3 - 5
\]
\[
f(x+h) - f(x) = (7x^3 + 21x^2h + 21xh^2 + 7h^3 - 5) - (7x^3 - 5) = 21x^2h + 21xh^2 + 7h^3
\]
\[
f'(x) = \lim_{h \to 0} \frac{21x^2h + 21xh^2 + 7h^3}{h} = \lim_{h \to 0} (21x^2 + 21xh + 7h^2) = 21x^2
\]
### 6. \( f(x) = -6x^3 + 9x^2 \)
\[
f(x+h) = -6(x+h)^3 + 9(x+h)^2 = -6(x^3 + 3x^2h + 3xh^2 + h^3) + 9(x^2 + 2xh + h^2)
\]
\[
= -6x^3 - 18x^2h - 18xh^2 - 6h^3 + 9x^2 + 18xh + 9h^2
\]
\[
f(x+h) - f(x) = (-6x^3 - 18x^2h - 18xh^2 - 6h^3 + 9x^2 + 18xh + 9h^2) - (-6x^3 + 9x^2) = -18x^2h + 18xh - 18xh^2 - 6h^3 + 9h^2
\]
\[
= -18x^2h + 18xh + 3h^2
\]
\[
f'(x) = \lim_{h \to 0} \frac{-18x^2h + 18xh + 3h^2}{h} = \lim_{h \to 0} (-18x^2 + 18x + 3h) = -18x^2 + 18x
\]
### 7. \( f(x) = x^3 + 6x^2 + 11x \)
\[
f(x+h) = (x+h)^3 + 6(x+h)^2 + 11(x+h) = (x^3 + 3x^2h + 3xh^2 + h^3) + 6(x^2 + 2xh + h^2) + 11x + 11h
\]
\[
= x^3 + 3x^2h + 3xh^2 + h^3 + 6x^2 + 12xh + 6h^2 + 11x + 11h
\]
\[
f(x+h) - f(x) = (x^3 + 3x^2h + 3xh^2 + h^3 + 6x^2 + 12xh + 6h^2 + 11x + 11h) - (x^3 + 6x^2 + 11x) = (3x^2 + 12x + 11)h + (3xh^2 + 6h^2 + h^3)
\]
\[
= (3x^2 + 12x + 11)h + (3x + 6)h^2 + h^3
\]
\[
f'(x) = \lim_{h \to 0} \frac{(3x^2 + 12x + 11)h + (3x + 6)h^2 + h^3}{h} = \lim_{h \to 0} (3x^2 + 12x + 11 + (3x + 6)h + h^2) = 3x^2 + 12x + 11
\]
### Resumen de las derivadas:
1. \( f'(x) = 6x \)
2. \( f'(x) = 6x^2 - 15 \)
3. \( f'(x) = -8x + 19 \)
4. \( f'(x) = 28 \)
5. \( f'(x) = 21x^2 \)
6. \( f'(x) = -18x^2 + 18x \)
7. \( f'(x) = 3x^2 + 12x + 11 \)
Quick Answer
1. \( f'(x) = 6x \)
2. \( f'(x) = 6x^2 - 15 \)
3. \( f'(x) = -8x + 19 \)
4. \( f'(x) = 28 \)
5. \( f'(x) = 21x^2 \)
6. \( f'(x) = -18x^2 + 18x \)
7. \( f'(x) = 3x^2 + 12x + 11 \)
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