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.

33197 questions

vote

1 answer

Solve given qus without using partial fraction method

$$z=f\left(x,y\right)=x^{2}\tan^{-1}\left(\frac{y}{x}\right)-y^{2}\tan^{-1}\left(\frac{x}{y}\right)$$ Prove that $$\frac{\partial^{2}f\left(x,y\right)}{\partial x\,\partial y}=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$$

derivatives

asked Aug 25 '14 at 18:30

Aakash

vote

1 answer

Finding the derivative of $y=12x^4\sqrt[3]{x^2}-2e^x+9$

Let $$ y=12x^4\sqrt[3]{x^2}-2e^x+9 $$ How can we find $y^\prime$?

derivatives

asked Aug 25 '14 at 01:25

Peep

vote

2 answers

Prove that $\lim_{x\to\infty}f''(x)=0$ if $\lim_{x\to\infty}f(x)=T$ and $\lim_{x\to\infty}f'''(x)=0$.

Suppose that $f$ is a real function such that $f(x) \to T$, where $T$ is a finite limit, and that $f''' \to 0$ as $x \to \infty$. Prove that $f''(x) \to 0$ as $x \to \infty$.

derivatives

asked Aug 17 '14 at 18:42

kammy

vote

3 answers

Understanding the logic behind particular derivative

I have $\frac{\partial (f(x) g(x))}{\partial x}$=$g(x) f'(x)+f(x) g'(x)$, I need to differentate this function with respect to x. $f(x)=(x+1) (x+2)^2 (x+3)^3 (x+4)^4$ However I do not see the logic using the product rule.

derivatives

asked Jul 28 '14 at 03:54

ALEXANDER

2,099

vote

3 answers

Derivative of $\sqrt{x^2+1}$

Ive been given this rule and asked to differentiate $\sqrt{x^2+1}$, however I am not sure what I am missing.It is said that if f is differentiable at x and f(x)>0. $\frac{d}{\text{dx}}$$\sqrt{f(x)}=\frac{f'(x)}{2 \sqrt{f(x)}}$ What I thought would…

derivatives

asked Jul 25 '14 at 20:39

ALEXANDER

2,099

vote

1 answer

Please help me check this derivative work - logistic regression algorithm related

I have $$ J_{\theta}(X) = - \frac 1 m \cdot \left[ y \cdot ln( h_{\theta} (X ) ) + ( 1 - y) \cdot ln ( 1 - h_{\theta}(X) ) \right] $$ I need $\frac d {d\theta} J_{\theta}(X)$. I tried many time, and here's my result $$ \frac d {d\theta}…

derivatives

asked Jul 22 '14 at 04:32

David S.

vote

1 answer

Find the equation of a line tangent at a specific point

I have to find an equation for the line tangent to the graph of $\large\frac {\sqrt{x}}{6x+5}$ at the point $(4,f(4))$, and write it out in the form of $y=mx+b$ Using the quotient rule I get.. $(6x+5)\frac12 x^{-{\frac12}} -…

derivatives

asked Jul 15 '14 at 03:31

jrounsav

vote

1 answer

Derivatives in the real world

Two row boats start at the same location, and start traveling apart along straight lines which meet at an angle of $\pi/3$. Boat A is traveling at a rate of $10$ miles per hour directly east, and boat $B$ is traveling at a rate of $16$ miles per…

derivatives

asked Jul 04 '14 at 01:28

Minu

vote

2 answers

calculate derivation in an equation

I need to calculate systematic error for $\tau$ in capacitor's charging formula($V_c(t)=V_s\left(1-e^{-t\over\tau}\right)$ ) I converted it to : $\tau=-{t \over \ln(1-{V_c \over V_s})}$ and continued by doing:…

derivatives

asked Nov 25 '11 at 11:45

Ariyan

vote

2 answers

How do I differentiate ${(e^e)}^x$?

I know how to differentiate $e^x$ (it's just $e^x$), but how do I differentiate ${(e^e)}^x$? any hints would be welcome.

derivatives

asked Jun 30 '14 at 14:14

user160325

vote

4 answers

Derivative of $(5x-2)^3$

How is the derivative of $(5x-2)^3$ equal to $15(5x-2)^2$ and not $3(5x-2)^2$. According to $\frac{df}{dx} = nx^{n-1}$, it has to be $3(5x-2)^2$ right. Please explain.

derivatives

asked Jun 26 '14 at 00:04

Minu

vote

1 answer

Showing that $\frac{d^m}{dx^m}[f(x)g(x)] + {{m+1}\choose k}\frac{d^{m+1+k}}{dx^{m+1+k}}f(x)\frac{d^k}{dx^k}g(x) = \frac{d^{m+1}}{dx^{m+1}}[f(x)g(x)]$

I need to algebraically, or using basic calculus, show that $$\displaystyle\frac{d^m}{dx^m}[f(x)g(x)] + {{m+1}\choose k}\frac{d^{m+1+k}}{dx^{m+1+k}}f(x)\frac{d^k}{dx^k}g(x) = \frac{d^{m+1}}{dx^{m+1}}[f(x)g(x)]$$ For reference, it's part of a proof…

derivatives

asked Jun 25 '14 at 21:26

Alec

4,094

vote

2 answers

Derivative of 1/sin x

I want to find out the derivative of 1/sin(x) without using the reciprocal rule. Let f(x) = 1/sin(x) Df/dx = (f(x+h) - f(x))/ h I keep getting 0 as the answer while the actual derivative according to the reciprocal rule is -cos(x)/sin^2(x) Can…

derivatives

asked Jun 24 '14 at 02:53

Minu

vote

1 answer

Calculating $\frac{d}{d(x^2)}f(x)$

There's a question I need to solve, which requires that I take the derivative of some function by the square of a variable, and I'm not sure how to do such a thing. For example: $\frac{dx}{d(x^2)}$ - I've tried $t=x^2$, $d\sqrt{t}/dt$ is easy enough…

derivatives

asked Jun 22 '14 at 17:00

user1999728

Prev 1 2 3

…

99 100 Next