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Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value)

Derivative of a function has a very natural geometric and physical interpretation: it corresponds to slope of the tangent line and to instantaneous velocity. In applications, it usually describes the rate of change of a physical variable.

Basic techniques used for computing the derivative of a given function are

It is useful to know the derivatives of elementary functions. This tag is intended for questions on the evaluation of derivatives.

Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

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Given that $f(x)=\frac{1}{x^n}$, show that $x f'(x)+n f'(x)=0$.

This exercise was in my math book and of course had no solution as it's a "show" type of question. I don't see how this could hold except for when $x=-n$. Given that $f(x)=\frac{1}{x^n}$, show that $x f'(x)+n f'(x)=0$.
O. J.
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Prove that $\frac{dy}{dx} = -\frac1{(1+x)^2}$ for given that $x\sqrt{1+y} + y\sqrt{1+x} = 0$

$$x\sqrt{1+y} + y\sqrt{1+x} = 0$$ Please tell me where I went wrong. Why I am not getting correct answer ?
Aakash Kumar
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Finding the second derivative of $x^x$

Find the second derivative $d^2y/dx^2$ when $y=x^x\:(x>0)$. $$y=x^x,\:\:(x\gt0)$$ \begin{align} \log y&=x\log x \\ \rm{Differentiating}&\:{\rm{with\:respect\:to\:}}x \end{align} \begin{align} \frac{1}{y}\frac{dy}{dx}&=1\cdot(\log…
Aakash Kumar
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Simplifying derivative result

I am doing the derivative of $$f(x) = \frac{x^2 -4x +3}{x^2-1}$$ So my result is the following $$f'(x) = \frac{4x^2 -8x +4}{(x^2-1)^2}$$ I am sure the answer is correct, but in my solutions book and In Wolfram Alpha they simplify until $$f'(x) =…
jsertx
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Partial Derivative of $xy^2+yz^2+xyz+x^2y^2z^2=5$

Someone can tell me what the Partial Derivative of $\frac{d^2z}{dy^2}$ of function $z(x,y)$ if it`s look like this: $$xy^2+yz^2+xyz+x^2y^2z^2=5$$ I try to solve the first derivative: $$\frac{dz}{dy}=(2yx+z^2z′+xzz′+2x^2yz^2z′)$$ but I am not sure if…
BAM
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prove or disprove by counter example

If $f$ is differentiable at $x=0$ and $\lim_{x\to0}{f(x)\over x}=3$ then $f(0)=0$ and $f'(0)=3 $. so after a few failed counter examples I decided to prove this since it also kinda seems to be true... this is where I got to: $f$ is differentiable…
Rubenz
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I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$.

I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$. My logic is I want to find whether is $f'(5)>0$ or $f'(5) < 0$. I need to use the chain rule $h'(x) = g'(f(x))f'(x)$ $g'(f(x)) =…
dagda1
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Inconsistencies with multiple differentiation methods

$$w=\sin x$$ $$\frac{dw}{dx} = \cos x$$ $$\therefore\frac{dx}{dw} = \frac{1}{\cos x}$$ Rearranging the initial relationship; $$x = \arcsin(w)$$ $$\therefore\frac{dx}{dw} = \frac{1}{(1-w^2)^{0.5}}$$ But, $$\frac{1}{\cos x} \neq…
StopReadingThisUsername
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using L'Hospital solve $\lim_{x \to \infty} x - x^{2}\ln(1 + \frac{1}{x})$

I can't get this to $ = \frac{0}{0}$ form so I can use l'Hospital rule $$\lim_{x \to \infty} x - x^{2}\ln\left(1 + \frac{1}{x}\right)$$ tips? [EDIT] $$\lim_{x \to 0} \frac{1}{x} - \frac{\ln(1 + x)}{x^{2}}$$ second term $$\lim_{x \to 0} \frac{\ln(1 +…
tomtom
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Understanding how this derivative was taken

I am pouring water into a conical cup 8cm tall and 6cm across the top. If the volume of the cup at time t is $V(t)$, how fast is the water level ($h$) rising in terms of $V'(t)$? The solution in the book is: Take the water volume, given…
naiveai
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Partial Derivative of $x^y$

$$\frac{\partial}{\partial x}x^y$$ $x^y=e^{ln(x)^y}=e^{y*ln(x)}$ $$\frac{\partial}{\partial x}e^{y\cdot ln(x)}=e^{y\cdot ln(x)}\cdot\frac{y}{x}=x^y\cdot\frac{y}{x}=x^{y-1}\cdot y$$ But the answer is different , where did I get it wrong?
gbox
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Find derivative using the limit definition for the function $f(x) = e^{2(x+1)}$

Use the limit definition of derivative to find $\frac{\mathrm d}{\mathrm dx}f(x)$ for the function $f(x)= e^{2(x+1)}$. I know it's going to be $2e^{2x+2}$. I can solve it normally, I just don't know how to solve it using the definition, please help.…
Tobias
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General question of derivatives and their inversions

If $\frac{df(x,y)}{dx} = a$, does $\frac{1}{a} = \frac{dx}{df(x,y)}$? Consider $f(x,y) = x^2y \Rightarrow \frac{df(x,y)}{dx} = 2xy \equiv a$, than $\frac{1}{a} = \frac{1}{2xy}$. Now calculate $\frac{dx}{df(x,y)} = \frac{dx}{d(x^2y)}$ via…
rtime
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Can $f''(x)$ exist if $f'(x)$ is undefined?

For example, the piecewise function $ f(x) = \begin{cases} \sqrt{1 - (x + 1)^2} &-2 \leq x \leq 0 \\ -\sqrt{1 - (x - 1)^2} &0 \leq x \leq 2 \end{cases} $ will, at $f(0)$, give $f'(0) = $ undefined (vertical tangent). Once deriving I can prove…
Avritt Rohwer
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Textual explanation of a derivative

In the book Structure and Interpretation of Computer Programs, there is an interesting example on how one might explore symbolic data in programming. They used the differentiation algorithm. That is, there is an algebraic expression (E.G. x+3) and…
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