The left hand derivative of the given function comes out to be 3a² + 3ah + h².
Deducing the Left Derivative:
The given function is,
f(x) = x³ + 2
⇒ f(a) = a³ + 2
The left hand limit is the definition of the left-hand derivative of f: f′⁻(x) = [tex]lim_{h- > 0}[/tex]f(x+h)f(x)h. F is said to be left-hand differentiable at x if the left-hand derivative exists.
Now, the formula for the left derivative of a function is given as,
f'(a)⁻ = [ f(a+h) - f(a) ] / [ (a+h) - a]
f'(a)⁻ = [ ((a+h)³ + 2) - (a³+2) ] / h
f'(a)⁻ = (a³ + 3a²h + 3ah² + h³ + 2 - a³ - 2) / h
f'(a)⁻ = (3a²h + 3ah² + h³) / h
f'(a)⁻ = h(3a² + 3ah + h²) / h
f'(a)⁻ = 3a² + 3ah + h²
Hence, the left derivative is 3a² + 3ah + h².
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