Finding Derivatives

Find the derivative of the function $y = 2x^{3}-5x^{2}+7x-12$.

Find the derivative of the function $y = x^{4}-3x^{2}+17$.

Find the derivative of the function $y = x-\frac{1}{2}x^{2}+\frac{1}{3}x^{3}-\frac{1}{4}x^{4}$.

Find the derivative of the function $y = x^{3}(x^{2}-1)^{2}$.

Find the derivative of the function $y = \frac{x-1}{x+1}$.

Find the derivative of the function $y = \frac{x}{1-x^{2}}$.

Find the derivative of the function $y = \frac{3x-1}{x^{5}}$.

Find the derivative of the function $y = \frac{x^{2}+x+1}{x^{2}-x+1}$.

Find the derivative of the function $y = \sqrt{x}$.

Find the derivative of the function $y = \sqrt[3]{x}$.

Find the derivative of the function $y = \sqrt{x\sqrt{x\sqrt{x}}}$.

Find the derivative of the function $y = e^{x}(x^{2}-2x+2)$.

Find the derivative of the function $y = x \sin x$.

Find the derivative of the function $y = x^{2} \sin x+2x \cos x-2 \sin x$.

Find the derivative of the function $y = x \ln x-x$.

Find the derivative of the function $y = \frac{x^{3}}{3} \ln x-\frac{1}{9}x^{3}$.

Find the derivative of the function $y = \sqrt{x}+\ln x-\frac{1}{\sqrt{x}}$.

Find the derivative of the function $y = \frac{1}{\sin x}$.

Find the derivative of the function $y = \frac{1}{\cos x}$.

Find the derivative of the function $y = \frac{1}{\ln x}$.

Find the derivative of the function $y = \frac{e^{x}-1}{e^{x}+1}$.

Find the derivative of the function $y = e^{x}(\sin x-\cos x)$.

Find the derivative of the function $y = (ax+b)^{n}$.

Find the derivative of the function $y = \sin^{3} x$.

Find the derivative of the function $y = (x^{2}-1)^{5}$.

Find the derivative of the function $y = \cos^{5} x$.

Find the derivative of the function $y = \sqrt{3x-5}$.

Find the derivative of the function $y = \sin 5x$.

Find the derivative of the function $y = \sqrt{a^{2}-x^{2}}$.

Find the derivative of the function $y = e^{-x}$.

Find the derivative of the function $y = \frac{x}{\sqrt{1-x^{2}}}$.

Find the derivative of the function $y = \ln \sin x$.

Find the derivative of the function $y = \frac{1}{(1+x^{2})\sqrt{1+x^{2}}}$.

Find the derivative of the function $y = \ln \operatorname {tg} x$.

Find the derivative of the function $y = x\sqrt{x^{2}+1}$.

Find the derivative of the function $y = e^{-x^{2}}$.

Find the derivative of the function $y = \sqrt{x+\sqrt{x}}$.

Find the derivative of the function $y = \ln (x^{3}+x^{2})$.

Find the derivative of the function $y = \sqrt[3]{(2x+1)^{2}}$.

Find the derivative of the function $y = \ln \left(x+\sqrt{x^{2}+a^{2}}\right)$.

Find the derivative of the function $y = \cos ax \cdot \sin bx$.

Find the derivative of the function $y = \ln \sqrt{\frac{1-x}{1+x}}$.

Find the derivative of the function $y = e^{ax} \cdot \cos bx$.

Find the derivative of the function $y = \ln \sqrt{\frac{1-\sin x}{1+\sin x}}$.

Find the derivative of the function $y = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$.

Find the derivative of the function $y = \ln \operatorname {tg}\left(\frac{x}{2}+\frac{\pi }{4}\right)$.

Find the derivative of the function $y = \frac{x^{4}}{4}\left((\ln x)^{2}-\frac{1}{2}\ln x+\frac{1}{8}\right)$.

Find the derivative of the function $y = \ln \frac{1+\sqrt{1-x^{2}}}{x}$.

Find the derivative of the function $y = \ln \frac{x^{2}+1+\sqrt{x^{4}+3x^{2}+1}}{x}$.

Find the derivative of the function $y = \operatorname {ctg} \pi x+\frac{\cos \pi x}{2 \sin^{3} \pi x}$.

Find the derivative of the function $y = e^{\sqrt[3]{x}}\left(\sqrt[3]{x^{2}}-2\sqrt[3]{x}+2\right)$.

Find the derivative of the function $y = (x+2a)\sqrt{x^{2}-a^{2}}+a^{2} \ln \left(x+\sqrt{x^{2}-a^{2}}\right)$.

Find the derivative of the function $y = \frac{\sin (2\ln x)-\cos (2\ln x)}{x^{2}}$.

Find the derivative of the function $y = \frac{1}{\sqrt{2}} \ln \left(\sqrt{2} \cdot \operatorname {tg} x+\sqrt{1+2\operatorname {tg}^{2} x}\right)$.

Find the derivative of the function $y = \ln \frac{1+\sqrt{x^{2}+1}}{x}-\sqrt{x^{2}+1}$.

Find the derivative of the function $y = \ln \frac{3-x^{2}}{2-x^{2}}$.

Find the derivative of the function $y = \frac{1}{2a}\left(\ln \frac{\sqrt{a^{2}+x^{2}}}{a+x}-\frac{a}{a+x}\right)$.

Find the derivative of the function $y = \ln \frac{1}{\sqrt{1-x^{4}}}$.

Find the derivative of the function $y = \operatorname {tg} x+\operatorname {tg}^{3} x+\frac{3}{5}\operatorname {tg}^{5} x+\frac{1}{7}\operatorname {tg}^{7} x$.

Find the derivative of the function $y = -\frac{\cos x}{2\sin^{2} x}+\frac{1}{2}\ln \operatorname {tg} \frac{x}{2}$.

Find the derivative of the function $y = \frac{1}{2}\operatorname {tg}^{2} x+\ln \cos x$.

Find the derivative of the function $y = \frac{\sin x}{2\cos^{2} x}-\frac{1}{2}\ln \operatorname {tg}\left(\frac{\pi }{4}-\frac{x}{2}\right)$.

Find the derivative of the function $y = \operatorname {tg} x \cdot \ln \cos x+\operatorname {tg} x-x$.

Find the derivative of the function $y = \ln \frac{\sqrt{a+be^{x}}-\sqrt{a}}{\sqrt{a+be^{x}}+\sqrt{a}}$.

Find the derivative of the function $y = \ln \frac{e^{x}+2+\sqrt{e^{2x}+4e^{x}+1}}{e^{x}+2-\sqrt{e^{2x}+4e^{x}+1}}$.

Find the derivative of the function $y = (x-2)\sqrt{1+e^{x}}-\ln \frac{\sqrt{1+e^{x}}-1}{\sqrt{1+e^{x}}+1}$.

Find the derivative of the function $y = \ln \operatorname {tg} \frac{x}{2}-\operatorname {ctg} x \cdot \ln (1+\sin x)-x$.

Find the derivative of the function $y = \ln \frac{a+b\cos nx+\sqrt{b^{2}-a^{2}}\sin nx}{a+b\cos nx}$.

Find the derivative of the function $y = (1-\sqrt{1-x})\ln x+2\sqrt{1-x}-2\ln (1+\sqrt{1-x})$.

Find the derivative of the function $y = \frac{x^{2}\ln x}{a+bx^{2}}-\frac{1}{b}\ln \sqrt{a+bx^{2}}$.

Find the derivative of the function $y = \sqrt{x}\ln \left(\sqrt{x}+\sqrt{x+a}\right)-\sqrt{x+a}$.

Find the derivative of the function $y = \frac{1}{a}\operatorname {arctg} \frac{x}{a}$.

Find the derivative of the function $y = \arcsin \frac{x}{a}$.

Find the derivative of the function $y = \arcsin \frac{1}{x}$.

Find the derivative of the function $y = (\arcsin x)^{2}$.

Find the derivative of the function $y = \arcsin \sin x$.

Find the derivative of the function $y = \arccos \frac{1}{x}$.

Find the derivative of the function $y = \sqrt{x^{2}-a^{2}}+a \arcsin \frac{a}{x}$.

Find the derivative of the function $y = \arccos \frac{2-x}{x\sqrt{2}}$.

Find the derivative of the function $y = \arcsin \sqrt{1-x^{2}}$.

Find the derivative of the function $y = \operatorname {arctg} \frac{x+1}{x-1}$; $x \neq 1$.

Find the derivative of the function $y = \frac{2}{\sqrt{3}} \operatorname {arctg} \frac{2x+1}{\sqrt{3}}$.

Find the derivative of the function $y = \arccos (3x-4x^{3})$.

Find the derivative of the function $y = \arcsin \frac{2x}{1+x^{2}}$.

Find the derivative of the function $y = \arcsin \frac{1-x^{2}}{1+x^{2}}$.

Find the derivative of the function $y = \frac{1}{\sqrt{3}} \operatorname {arctg} \frac{x\sqrt{3}}{1-x^{2}}$.

Find the derivative of the function $y = \frac{1}{\sqrt{ab}} \operatorname {arctg}\left(\sqrt{\frac{b}{a}} \operatorname {tg} x\right)$.

Find the derivative of the function $y = \frac{2}{3} \operatorname {arctg} x+\frac{1}{3} \operatorname {arctg} \frac{x}{1-x^{2}}$.

Find the derivative of the function $y = \frac{1}{4\sqrt{2}} \ln \frac{x^{2}+x\sqrt{2}+1}{x^{2}-x\sqrt{2}+1}+\frac{1}{2\sqrt{2}} \operatorname {arctg} \frac{x\sqrt{2}}{1-x^{2}}$.

Find the derivative of the function $y = \frac{2}{\sqrt{a^{2}-b^{2}}} \operatorname {arctg}\left(\sqrt{\frac{a-b}{a+b}} \operatorname {tg} \frac{x}{2}\right)$; $a > b \geq 0$.

Find the derivative of the function $y = \frac{2}{3} \operatorname {arctg}\left(\frac{5}{3} \operatorname {tg} \frac{x}{2}+\frac{4}{3}\right)$.

Find the derivative of the function $y = \frac{x}{\sqrt{1-x^{2}}} \arcsin x+\ln \sqrt{1-x^{2}}$.

Find the derivative of the function $y = \sqrt{ax-x^{2}}-a \operatorname {arctg} \sqrt{\frac{a-x}{x}}$.

Find the derivative of the function $y = \ln \frac{\sqrt{x^{2}-x+1}}{x}-\frac{\ln (x^{2}-x+1)}{x}+\sqrt{3} \operatorname {arctg} \frac{2x-1}{\sqrt{3}}$.

Find the derivative of the function $y = \sqrt{x}-\sqrt{1-x} \cdot \arcsin \sqrt{x}$.

Find the derivative of the function $y = \frac{3x+2}{4x^{2}}\sqrt{x-1}+\frac{3}{4} \operatorname {arctg} \sqrt{x-1}$.

Find the derivative of the function $y = \ln \frac{\sqrt{1+x^{2}}}{x}-\frac{1}{x^{3}} \operatorname {arctg} x-\frac{1}{2x^{2}}$.

Find the derivative of the function $y = \sqrt{(a-x)(x-b)}-(a-b) \operatorname {arctg} \sqrt{\frac{a-x}{x-b}}$.

Find the derivative of the function $y = \arcsin (\sin x-\cos x)+\ln \left(\sin x+\cos x+\sqrt{\sin 2x}\right)$.

Find the derivative of the function $y = 2\operatorname {arctg} \frac{\sqrt{2\operatorname {tg} x}}{1-\operatorname {tg} x}+\ln \frac{1+\sqrt{2\operatorname {tg} x}+\operatorname {tg} x}{1-\sqrt{2\operatorname {tg} x}+\operatorname {tg} x}$.

Find the derivative of the function $y = \ln \left(\operatorname {tg} x+\sqrt{2+\operatorname {tg}^{2} x}\right)+\operatorname {arctg} \frac{\operatorname {tg} x}{\sqrt{2+\operatorname {tg}^{2} x}}$.

Find the derivative of the function $y = x \operatorname {arctg}(1+\sqrt{x})+\ln (x+2\sqrt{x}+x)-\sqrt{x}$.

Find the derivative of the function $y = x \arcsin \sqrt{\frac{x}{x+1}}+\operatorname {arctg} \sqrt{x}-\sqrt{x}$.

Find the derivative of the function $y = \frac{\arcsin \sqrt{x}}{\sqrt{x}}+\ln \sqrt{\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}}$.

Find the derivative of the function $y = (x^{2}+1)(\operatorname {arctg} x)^{2}-2x \cdot \operatorname {arctg} x+\ln (1+x^{2})$.

Find the derivative of the function $y = \ln \frac{\sqrt[4]{1+x^{4}}+x}{\sqrt[4]{1+x^{4}}-x}-2\operatorname {arctg} \frac{\sqrt[4]{1+x^{4}}}{x}$.

Find the derivative of the function $y = x^{x}$.

Find the derivative of the function $y = x^{\sin x}$.

Find the derivative of the function $y = \left|x\right|$.

Find the derivative of the function $y = x^{2} \sin \frac{1}{x}$.

From the equality $1+x+x^{2}+\ldots +x^{n} = \frac{x^{n+1}-1}{x-1}$, find a formula for the sum $1+2x+3x^{2}+\ldots +nx^{n-1}$.

From the equality

find a formula for $\sin x+2\sin 2x+\ldots +n\sin nx$.

From the equality

obtain a formula for the sum

From the formula

derive the formula

Deduce from this that

Derive directly from the definition of the derivative the derivatives of the functions: $\arcsin x$; $\operatorname {arctg} x$ (see problems 948 and 949).

Verify that if $xy = 1$, then

If $x\sqrt{1-y^{4}} = y\sqrt{2}$, then

Prove this.