Chapter 3: Rules for Finding DerivativesContent check — Calculus (Guichard), Chapter 3: Rules for Finding Derivatives

Find the derivatives of the given functions.

${\displaystyle x^{100}}$

Find the derivatives of the given functions.

${\displaystyle x^{-100}}$

Find the derivatives of the given functions.

${\displaystyle \frac{1}{x^5}}$

Find the derivatives of the given functions.

${\displaystyle x^{\pi }}$

Find the derivatives of the given functions.

${\displaystyle x^{3/4}}$

Find the derivatives of the given functions.

${\displaystyle x^{-9/7}}$

Find the derivatives of the functions.

${\displaystyle 5x^3+12x^2-15}$

Find the derivatives of the functions.

${\displaystyle -4x^5+3x^2-5/x^2}$

Find the derivatives of the functions.

${\displaystyle 5(-3x^2+5x+1)}$

Find the derivatives of the functions.

$f(x)+g(x)$⁠, where ${\displaystyle f(x)=x^2-3x+2}$ and ${\displaystyle g(x)=2x^3-5x}$

Find the derivatives of the functions.

${\displaystyle (x+1)(x^2+2x-3)}$

Find the derivatives of the functions.

${\displaystyle \sqrt{625-x^2}+3x^3+12}$ (See section 2.1.)

Find an equation for the tangent line to ${\displaystyle f(x)=x^3/4-1/x}$ at $x=-2$⁠.

Find an equation for the tangent line to ${\displaystyle f(x)=3x^2-{\pi }^3}$ at $x=4$⁠.

Suppose the position of an object at time $t$ is given by ${\displaystyle f(t)=-49t^2/10+5t+10}$⁠. Find a function giving the speed of the object at time $t$⁠. The acceleration of an object is the rate at which its speed is changing, which means it is given by the derivative of the speed function. Find the acceleration of the object at time $t$⁠.

The general polynomial $P$ of degree $n$ in the variable $x$ has the form ${\displaystyle P(x)=}$ ${\displaystyle \sum _{k=0}^n{a}_k{x}^k=}$ ${\displaystyle a_0+}$ ${\displaystyle a_{1}x+}$ ${\displaystyle \dots +}$ ${\displaystyle a_n{x}^n}$⁠. What is the derivative (with respect to $x$⁠) of $P$⁠?

Find a cubic polynomial whose graph has horizontal tangents at $(-2,5)$ and $(2,3)$⁠.

Find the derivatives of the functions using the product rule.

${\displaystyle x^3(x^3-5x+10)}$

Find the derivatives of the functions using the product rule.

${\displaystyle (x^2+}$ ${\displaystyle 5x-}$ ${\displaystyle 3)(x^5-}$ ${\displaystyle 6x^3+}$ ${\displaystyle 3x^2-}$ ${\displaystyle 7x+}$ ${\displaystyle 1)}$

Find the derivatives of the functions using the product rule.

${\displaystyle \sqrt{x}\sqrt{625-x^2}}$

Find the derivatives of the functions using the product rule.

${\displaystyle \frac{\sqrt{625-x^2}}{x^{20}}}$

Use the product rule to compute the derivative of ${\displaystyle f(x)={(2x-3)}^2}$⁠. Sketch the function. Find an equation of the tangent line to the curve at $x=2$⁠. Sketch the tangent line at $x=2$⁠.

Find the derivatives of the functions using the quotient rule.

${\displaystyle \frac{x^3}{x^3-5x+10}}$

Find the derivatives of the functions using the quotient rule.

${\displaystyle \frac{x^2+5x-3}{x^5-6x^3+3x^2-7x+1}}$

Find the derivatives of the functions using the quotient rule.

${\displaystyle \frac{\sqrt{x}}{\sqrt{625-x^2}}}$

Find the derivatives of the functions using the quotient rule.

${\displaystyle \frac{\sqrt{625-x^2}}{x^{20}}}$

Find an equation for the tangent line to ${\displaystyle f(x)=(x^2-4)/(5-x)}$ at $x=3$⁠.

Find an equation for the tangent line to ${\displaystyle f(x)=(x-2)/(x^3+4x-1)}$ at $x=1$⁠.

The curve ${\displaystyle y=1/(1+x^2)}$ is an example of a class of curves each of which is called a witch of Agnesi. Sketch the curve and find the tangent line to the curve at $x=5$⁠. (The word witch here is a mistranslation of the original Italian, as described at http://mathworld.wolfram.com/WitchofAgnesi.html and http://witchofagnesi.org/.

If $f^{\prime }(4)=5$⁠, $g^{\prime }(4)=12$⁠, $(fg)(4)=f(4)g(4)=2$⁠, and $g(4)=6$⁠, compute $f(4)$ and ${\displaystyle \frac{d}{dx}\frac{f}{g}}$ at 4.

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle x^4-3x^3+(1/2)x^2+7x-\pi }$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle x^3-2x^2+4\sqrt{x}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {(x^2+1)}^3}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle x\sqrt{169-x^2}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle (x^2-4x+5)\sqrt{25-x^2}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \sqrt{r^2-x^2}}$⁠, $r$ is a constant

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \sqrt{1+x^4}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {\displaystyle \frac{1}{\sqrt{5-\sqrt{x}}}}}$⁠.

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {(1+3x)}^2}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{x^2+x+1}{1-x}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{\sqrt{25-x^2}}{x}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \sqrt{\frac{169}{x}-x}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \sqrt{x^3-x^2-(1/x)}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle 100/{(100-x^2)}^{3/2}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \sqrt[3]{x+x^3}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \sqrt{{(x^2+1)}^2+\sqrt{1+{(x^2+1)}^2}}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {(x+8)}^5}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {(4-x)}^3}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {(x^2+5)}^3}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {(6-2x^2)}^3}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {(1-4x^3)}^{-2}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle 5(x+1-1/x)}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle 4{(2x^2-x+3)}^{-2}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{1}{1+1/x}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{-3}{4x^2-2x+1}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle (x^2+1)(5-2x)/2}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle (3x^2+1){(2x-4)}^3}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{x+1}{x-1}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{x^2-1}{x^2+1}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{(x-1)(x-2)}{x-3}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{2x^{-1}-x^{-2}}{3x^{-1}-4x^{-2}}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle 3(x^2+1)(2x^2-1)(2x+3)}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle \frac{1}{(2x+1)(x-3)}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {({(2x+1)}^{-1}+3)}^{-1}}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

${\displaystyle {(2x+1)}^3{(x^2+1)}^2}$

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

Find an equation for the tangent line to ${\displaystyle f(x)=}$ ${\displaystyle {(x-2)}^{1/3}/{(x^3+4x-1)}^2}$ at $x=1$⁠.

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

Find an equation for the tangent line to ${\displaystyle y=9x^{-2}}$ at $(\mathrm{3,1})$⁠.

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

Find an equation for the tangent line to ${\displaystyle (x^2-4x+5)\sqrt{25-x^2}}$ at $(\mathrm{3,8})$⁠.

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

Find an equation for the tangent line to ${\displaystyle {\displaystyle \frac{(x^2+x+1)}{(1-x)}}}$ at $(2,-7)$⁠.

Find the derivatives of the functions. For extra practice, and to check your answers, do some of these in more than one way if possible.

Find an equation for the tangent line to ${\displaystyle \sqrt{{(x^2+1)}^2+\sqrt{1+{(x^2+1)}^2}}}$ at ${\displaystyle (1,\sqrt{4+\sqrt{5}})}$⁠.