Find the derivative of a function with new variable

You calculated the derivative of the composite function $x\mapsto f(x^2+5)$, and evaluated it at $x$, rather than calculating the derivative of the function $t\mapsto f(t)$, and evaluating it at $x^2+5$. Note that by definition $$ \left(x\mapsto f(x^2+5)\right)'(x)=\lim_{h\to0}\frac{f((x+h)^2+5)-f(x^2+5)}{h} \, , $$ whereas $$ (t\mapsto f(t))'(x^2+5)=\lim_{h\to0}\frac{f(t+h)-f(t)}{h}\bigg\rvert_{t=x^2+5}=\lim_{h\to0}\frac{f(x^2+5+h)-f(x^2+5)}{h} \, . $$

As a method to proceed: $f(g(x))’=g’(x)f’(g(x))$. Set $g(x)=x^2+5$. So, $g’(x)=2x$. For $x$ non zero, $f’(x^2+5)=\frac{f(g(x))}{2x}$