Okay, let's find the derivative of the function [tex]\( f(x) = \frac{3}{2} x^2 + 4x - 5 \)[/tex] using the definition of the derivative.
The definition of the derivative, [tex]\( f'(x) \)[/tex], at any point [tex]\( x \)[/tex] is given by the limit of the difference quotient:
[tex]\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \][/tex]
### Step 1: Compute [tex]\( f(x+h) \)[/tex]
First, substitute [tex]\( x + h \)[/tex] into the function:
[tex]\[ f(x + h) = \frac{3}{2} (x + h)^2 + 4 (x + h) - 5 \][/tex]
Now, expand [tex]\( (x + h)^2 \)[/tex]:
[tex]\[ (x + h)^2 = x^2 + 2xh + h^2 \][/tex]
Thus, we can rewrite [tex]\( f(x + h) \)[/tex]:
[tex]\[
f(x + h) = \frac{3}{2} (x^2 + 2xh + h^2) + 4(x + h) - 5
\][/tex]
Distribute and combine like terms:
[tex]\[
f(x + h) = \frac{3}{2} x^2 + 3xh + \frac{3}{2} h^2 + 4x + 4h - 5
\][/tex]
### Step 2: Form the Difference Quotient
Now form the difference quotient [tex]\( \frac{f(x+h) - f(x)}{h} \)[/tex]:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{\left( \frac{3}{2} x^2 + 3xh + \frac{3}{2} h^2 + 4x + 4h - 5 \right) - \left( \frac{3}{2} x^2 + 4x - 5 \right)}{h}
\][/tex]
Subtract [tex]\( f(x) \)[/tex] from [tex]\( f(x+h) \)[/tex]:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{ \left( \frac{3}{2} x^2 + 3xh + \frac{3}{2} h^2 + 4x + 4h - 5 \right) - \left( \frac{3}{2} x^2 + 4x - 5 \right)}{h}
\][/tex]
Cancel out the common terms [tex]\( \frac{3}{2} x^2 \)[/tex], [tex]\( 4x \)[/tex], and [tex]\( -5 \)[/tex]:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{3xh + \frac{3}{2} h^2 + 4h}{h}
\][/tex]
### Step 3: Simplify and Take the Limit
Factor out [tex]\( h \)[/tex] in the numerator:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{h(3x + \frac{3}{2} h + 4)}{h}
\][/tex]
Cancel the [tex]\( h \)[/tex] in the numerator and denominator:
[tex]\[
\frac{f(x+h) - f(x)}{h} = 3x + \frac{3}{2} h + 4
\][/tex]
Take the limit as [tex]\( h \)[/tex] approaches 0:
[tex]\[
f'(x) = \lim_{{h \to 0}} (3x + \frac{3}{2} h + 4) = 3x + 4
\][/tex]
Thus, the derivative of the function [tex]\( f(x) = \frac{3}{2} x^2 + 4 x - 5 \)[/tex] is:
[tex]\[
f'(x) = 3x + 4
\][/tex]
