Giải bài tập sgk toán lớp 12 bài số 2 trang 61 Hàm số lũy thừa
Tìm các đạo hàm của các hàm số:
a) \(y= \left ( 2x^{2} -x+1\right )^{\frac{1}{3}}\);
b) \(y= \left ( 4-x-x^{2}\right )^{\frac{1}{4}}\);
c) \(y= \left ( 3x+1\right )^{\frac{\pi }{2}}\);
d) \(y= \left ( 5-x\right )^{\sqrt{3}}\).
a) \(y^{'}=\frac{1}{3}\left ( 2x^{2} -x+1\right )^{'}\left (2x^{2}-x+1 \right )^{\frac{1}{3}-1}\)= \(\frac{\left ( 4x-1\right )\left ( 2x^{2}-x+1 \right )^{\frac{-2}{3}}}{3}\).
b) \(y^{'}=\frac{1}{4}\left ( 4-x-x^{2} \right )^{'}\left ( 4-x-x^{2} \right )^{\frac{1}{4}-1}\)= \(\frac{1}{4}\left ( -2x-1 \right )\left ( 4-x-x^{2} \right )^{\frac{-3}{4}}\).
c) \(y^{'}\)= \(\frac{\pi }{2}\left ( 3x+1 \right )^{'}\left ( 3x+1 \right )^{\frac{\pi }{2}-1}\)= \(\frac{3\pi }{2}\left ( 3x+1 \right )^{\frac{\pi }{2}-1}\).
d) \(y^{'}\)= \(\sqrt{3}\left ( 5-x \right )^{'}\left ( 5-x \right )^{\sqrt{3}-1}\)= \(-\sqrt{3}\left ( 5-x \right )^{\sqrt{3}-1}\).
Tìm các đạo hàm của các hàm số:
a) \(y= \left ( 2x^{2} -x+1\right )^{\frac{1}{3}}\);
b) \(y= \left ( 4-x-x^{2}\right )^{\frac{1}{4}}\);
c) \(y= \left ( 3x+1\right )^{\frac{\pi }{2}}\);
d) \(y= \left ( 5-x\right )^{\sqrt{3}}\).
Lời giải bài tập chi tiết
a) \(y^{'}=\frac{1}{3}\left ( 2x^{2} -x+1\right )^{'}\left (2x^{2}-x+1 \right )^{\frac{1}{3}-1}\)= \(\frac{\left ( 4x-1\right )\left ( 2x^{2}-x+1 \right )^{\frac{-2}{3}}}{3}\).
b) \(y^{'}=\frac{1}{4}\left ( 4-x-x^{2} \right )^{'}\left ( 4-x-x^{2} \right )^{\frac{1}{4}-1}\)= \(\frac{1}{4}\left ( -2x-1 \right )\left ( 4-x-x^{2} \right )^{\frac{-3}{4}}\).
c) \(y^{'}\)= \(\frac{\pi }{2}\left ( 3x+1 \right )^{'}\left ( 3x+1 \right )^{\frac{\pi }{2}-1}\)= \(\frac{3\pi }{2}\left ( 3x+1 \right )^{\frac{\pi }{2}-1}\).
d) \(y^{'}\)= \(\sqrt{3}\left ( 5-x \right )^{'}\left ( 5-x \right )^{\sqrt{3}-1}\)= \(-\sqrt{3}\left ( 5-x \right )^{\sqrt{3}-1}\).