Vaughan Sanders
03/14/2024 · Junior High School
Calculen la derivada por definición de cada una de las siguientes funciones en el punto que se indica en cada caso. \( \begin{array}{ll}\text { a) } f(x)=4 x+3 \text {, en } x=5 & \text { d) } f(x)=x^{2}-2 x \text {, en } x=-2 \\ \text { b) } f(x)=5 x^{3}+1 \text {, en } x=1 & \text { e) } f(x)=2 x^{2}-3 x \text {, en } x=3\end{array} \)
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Step-by-step Solution
Para calcular la derivada de una función en un punto utilizando la definición de derivada, utilizamos la siguiente fórmula:
\[
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
\]
Ahora, aplicaremos esta definición a cada una de las funciones y puntos indicados.
### a) \( f(x) = 4x + 3 \), en \( x = 5 \)
1. Calcular \( f(5) \):
\[
f(5) = 4(5) + 3 = 20 + 3 = 23
\]
2. Calcular \( f(5+h) \):
\[
f(5+h) = 4(5+h) + 3 = 20 + 4h + 3 = 23 + 4h
\]
3. Sustituir en la definición de la derivada:
\[
f'(5) = \lim_{h \to 0} \frac{(23 + 4h) - 23}{h} = \lim_{h \to 0} \frac{4h}{h} = \lim_{h \to 0} 4 = 4
\]
### b) \( f(x) = 5x^3 + 1 \), en \( x = 1 \)
1. Calcular \( f(1) \):
\[
f(1) = 5(1)^3 + 1 = 5 + 1 = 6
\]
2. Calcular \( f(1+h) \):
\[
f(1+h) = 5(1+h)^3 + 1 = 5(1 + 3h + 3h^2 + h^3) + 1 = 5 + 15h + 15h^2 + 5h^3 + 1 = 6 + 15h + 15h^2 + 5h^3
\]
3. Sustituir en la definición de la derivada:
\[
f'(1) = \lim_{h \to 0} \frac{(6 + 15h + 15h^2 + 5h^3) - 6}{h} = \lim_{h \to 0} \frac{15h + 15h^2 + 5h^3}{h} = \lim_{h \to 0} (15 + 15h + 5h^2) = 15
\]
### d) \( f(x) = x^2 - 2x \), en \( x = -2 \)
1. Calcular \( f(-2) \):
\[
f(-2) = (-2)^2 - 2(-2) = 4 + 4 = 8
\]
2. Calcular \( f(-2+h) \):
\[
f(-2+h) = (-2+h)^2 - 2(-2+h) = (4 - 4h + h^2) + 4 - 2h = 8 - 6h + h^2
\]
3. Sustituir en la definición de la derivada:
\[
f'(-2) = \lim_{h \to 0} \frac{(8 - 6h + h^2) - 8}{h} = \lim_{h \to 0} \frac{-6h + h^2}{h} = \lim_{h \to 0} (-6 + h) = -6
\]
### e) \( f(x) = 2x^2 - 3x \), en \( x = 3 \)
1. Calcular \( f(3) \):
\[
f(3) = 2(3)^2 - 3(3) = 2(9) - 9 = 18 - 9 = 9
\]
2. Calcular \( f(3+h) \):
\[
f(3+h) = 2(3+h)^2 - 3(3+h) = 2(9 + 6h + h^2) - 9 - 3h = 18 + 12h + 2h^2 - 9 - 3h = 9 + 9h + 2h^2
\]
3. Sustituir en la definición de la derivada:
\[
f'(3) = \lim_{h \to 0} \
Quick Answer
a) \( f'(5) = 4 \)
b) \( f'(1) = 15 \)
d) \( f'(-2) = -6 \)
e) \( f'(3) = 9 \)
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