If _0^xf(t),dt=x^2+_x^1t^2f(t),dt, then f'(1/2) is:

Question

If $\displaystyle\int_0^xf(t)\,dt=x^2+\int_x^1t^2f(t)\,dt$, then $f'(1/2)$ is:

Accepted Answer

Differentiating both sides with respect to $x$:
$$f(x)=2x-x^2f(x)\Rightarrow f(x)=\frac{2x}{1+x^2}$$

$$f'(x)=\frac{2(1-x^2)}{(1+x^2)^2}$$

$$f'\left(\frac12\right)=\frac{2\left(1-\frac14\right)}{\left(1+\frac14\right)^2}=\frac{\frac32}{\frac{25}{16}}=\frac32\cdot\frac{16}{25}=\frac{24}{25}$$

Solutions · 1