# Derivatives calculus questions and answers Recent questions in Derivatives
Derivatives
ANSWERED ### If $$\displaystyle{y}{\left({x}\right)}={\sin{{u}}}$$, then y' is Option 1:$$\displaystyle-{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}{\cos{{u}}}$$ Option 2:$$\displaystyle{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}{\cos{{u}}}$$ Option 3:$$\displaystyle{\cos{{u}}}$$ Option 4:$$\displaystyle-{\cos{{u}}}$$

Derivatives
ANSWERED ### Solve the given differential equation. $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{y}}}{{{x}}}}$$

Derivatives
ANSWERED Derivatives

### Verify that the hypotheses of Rolle’s Theorem are satisfied for f(x) = $$1\over6$$x - $$\sqrt {x}$$ on the interval [0,36], and find the value of c in this interval that satisfies the conclusion of the theorem.

Derivatives
ANSWERED ### Second derivatives Find $$\displaystyle{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}$$ $$\displaystyle{x}+{y}={\sin{{y}}}$$

Derivatives
ANSWERED ### Find the derivatives $$\displaystyle{y}={\left({1}-{4}{x}+{7}{x}^{{{5}}}\right)}^{{{30}}}$$

Derivatives
ANSWERED ### Find derivatives of $$\displaystyle{r}={\frac{{{12}}}{{{0}}}}-{\frac{{{4}}}{{{0}^{{{3}}}}}}+{\frac{{{1}}}{{{0}^{{{4}}}}}}$$

Derivatives
ANSWERED ### Solve the derivatives. $$\displaystyle{v}={\left({z}^{{{4}}}-{2}{z}+{1}\right)}^{{{\frac{{{3}}}{{{2}}}}}}$$

Derivatives
ANSWERED ### Find the indicated derivatives. $$\displaystyle{\frac{{{\left.{d}{z}\right.}}}{{{d}{w}}}}\ {\quad\text{if}\quad}\ {z}={\frac{{{1}}}{{\sqrt{{{w}^{{{2}}}-{1}}}}}}$$

Derivatives
ANSWERED ### Compute the derivatives indicated. $$\displaystyle{f{{\left({x},{y}\right)}}}={3}{x}^{{{2}}}{y}-{6}{x}{y}^{{{4}}},{\frac{{\partial^{{{2}}}{f}}}{{\partial{x}^{{{2}}}}}}\ {\quad\text{and}\quad}\ {\frac{{\partial^{{{2}}}{f}}}{{\partial{y}^{{{2}}}}}}$$

Derivatives
ANSWERED ### Find the indicated derivatives. $$\displaystyle{\frac{{{d}}}{{{d}{u}}}}{\left({8}{u}^{{{\frac{{{3}}}{{{4}}}}}}+{4}{u}^{{-{\frac{{{1}}}{{{4}}}}}}\right)}$$

Derivatives
ANSWERED ### Use the rules for derivatives to find the derivatives below. $$\displaystyle{y}=\sqrt{{{x}}}-{x}^{{{2}}}$$; use the Power Rule

Derivatives
ANSWERED ### Write the first-order linear differential equation in standard form?

Derivatives
ANSWERED ### Find the derivatives of the following function. $$\displaystyle{f{{\left({x}\right)}}}={8}{x}^{{{5}}}+{2}{x}^{{{4}}}-{13}{x}^{{{3}}}$$ f'(x)=

Derivatives
ANSWERED ### Use the quotient rule and the derivatives of the sine and cosine functions to prove that $$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\cot{{x}}}\right)}=-{{\csc}^{{{2}}}{x}}$$.

Derivatives
ANSWERED ### Compute the derivatives of the functions $$\displaystyle{g{{\left({x}\right)}}}={\left(\sqrt{{{x}}}+{x}^{{{2}}}\right)}{\left({x}^{{{3}}}+{x}+\sqrt{{{2}}}\right)}$$

Derivatives
ANSWERED ### If y(x)=uv, then y' is Option 1:$$\displaystyle{u}{\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}}-{v}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}$$ Option 2:$$\displaystyle{u}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}+{v}{\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}}$$ Option 3:$$\displaystyle{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}\cdot{\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}}$$ Option 4:$$\displaystyle{u}{\frac{{{d}{v}}}{{{\left.{d}{x}\right.}}}}+{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}{v}$$

Derivatives
ANSWERED ### Differentiate the function. $$\displaystyle{G}{\left({x}\right)}=\sqrt{{x}^{{{6}}}+{6}{x}}$$ G'(x)=

Derivatives
ANSWERED ### With respect to x, what is the derivative of: $$\displaystyle\sqrt{u}^{{{2}}}$$ Option 1: $$\displaystyle{u}^{{{\frac{{{2}}}{{{5}}}}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}$$ Option 2: $$\displaystyle-{\frac{{{5}}}{{{3}}}}{u}^{{{\frac{{-{3}}}{{{5}}}}}}$$ Option 3: $$\displaystyle{\frac{{{2}}}{{{5}}}}{u}^{{{\frac{{-{3}}}{{{5}}}}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}$$ Option 4: $$\displaystyle{\frac{{{5}}}{{{7}}}}{u}^{{{\frac{{{7}}}{{{5}}}}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}$$
ANSWERED 