Suppose a differentiable function f (x) satisfies the identity f(x...

$∵ \lim_{x \to 0}\dfrac{f\left(x\right)}{x} = 1 ⇒ f ' \left(0\right) = 1$

$f\left(x + y\right) = f\left(x\right) + f\left(y\right) + x{y}^{2} + {x}^{2}y$

Differentiate w.r.t. $x$ keeping $y$ constant

$f ' \left(x + y\right) = f ' \left(x\right) + 0 + {y}^{2} + 2xy$

put $y = - x$

$f ' \left(0\right) = f ' \left(x\right) + {x}^{2} - 2{x}^{2}$

$1 = f ' \left(x\right) - {x}^{2}$

$f ' \left(x\right) = 1 + {x}^{2}$

$f ' \left(3\right) = 10$.