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

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2 answers

A problem on second order differentiation

If $y=\sin x$, then find the value of $$\frac{d^2(\cos^7 x)}{dy^2}$$ I have no idea on how to proceed in this problem. Please help.

derivatives

asked Nov 02 '13 at 14:13

Tejas

2,082

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1 answer

Proving two functions with independent variables are constant when inside an equation with isolated variables

So I've got: $\frac{1}{r}\frac{du}{dr}=\frac{1}{2\mu}\frac{dp}{dx}$ as an equation in a fluid mechanics book. The book stipulates that if we have a function that only depends on r ont he left side and that only depends on x on the right side ($\mu$…

derivatives

asked Oct 31 '13 at 15:42

user104521

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1 answer

Derivative of composite function with ln and square root

I need to find $f(x)'$ while $f(x) = ln(x+\sqrt{a^2+x^2})$ I'm having: $f(x)'=\frac{1}{(x+\sqrt{a^2+x^2})}\cdot\left(1+\frac{2x}{2\sqrt{a^2+x^2}}\right)$ ... can't simplify. I should get $\frac{1}{\sqrt{a^2+x^2}}$. Any suggestions greatly…

derivatives

asked Oct 29 '13 at 23:11

Jan Vorisek

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3 answers

Derivative Arc-tangens

We have to show that $(\arctan(x))' = \frac{1}{1+x^2}$, derived with the chain rule. The hint given is that we should start with deriving $\tan(\arctan(x)) = x$; I am not sure though how this is helpful, since the derivative of $\arctan(x)$ is what…

derivatives

asked Oct 29 '13 at 13:37

TestGuest

1,053

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3 answers

Find all values of $x$ at which the tangent line to the given curve has intercept $ y= 2$

Find all values of $x$ at which the tangent line to the given curve has intercept $y = 2$ I am confused about the $y$-intercept $2$ the function $$f(x) = \frac{(2x + 5)}{(x + 2)}$$ The derivative is $$f'(x) = \frac{- 1}{(x^2 + 4x + 4)}$$ Now what…

derivatives

asked Oct 26 '13 at 14:11

DevX

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1 answer

Value of $x$ where the tangent line is horizontal

Find the value of $x$ where the tangent line to the curve $y=2xe^{-x}$ is horizontal. I need your help please...

derivatives

asked Oct 15 '13 at 11:05

Alice1997

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1 answer

Just Require Some Checking For My Differential

If $f(x)=\cos(\sin(\cos(x)))$, $f'(x)=\sin(\sin(\cos(x)))\cos(\cos(x))\sin(x)$. Tried using Wolfram but it couldn't compute my integral. Am I correct?

derivatives

asked Oct 13 '13 at 15:41

Vladimir Nabokov

1,003

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1 answer

Verify a solution - Chain Rule

I've submitted a homework assignment online where it is then marked by a program and a score is instantly given back. I was 100% correct in all solutions besides this one. Use the Chain Rule to find $\displaystyle \frac{dw}{dt}$. $$ w =…

derivatives

asked Oct 11 '13 at 16:48

user100299

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Differentiation of function with chain rule

the following expression is part of a function I have to differentiate: $y = \tan^3(5x^4-7)$ I tried using the chain-rule, so: $ y' = 3\tan^2(5x^4-7)\cdot(20x^3)$ is this correct?

derivatives

asked Oct 11 '13 at 10:19

Vazrael

2,281

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0 answers

Zero in denominator while using logarithmic differentiation

I am presented with the following task: "Find $f'(1)$ using logarithmic differentiation, given that $$f(x) = \frac{(x^2-1)(x^2-2)(x^2-4)}{(x^2+1)(x^2+2)(x^2+3)}$$" I have done the following: $$\ln f(x) = \ln…

derivatives

asked Oct 10 '13 at 16:02

Andrew Thompson

4,431

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1 answer

How to determine concavity?

For the function $f(x)=x^3-9x^2$, how do I determine concavity? I know that for $f'(x) < 0$ the function is concave up when $f''(x)>0$, and concave down when $f''(x) <0$, etc... $f'(x) = 3x^2-18x$, $f''(x) = 6x-18$. Do I need to evaluate at a…

derivatives

asked Oct 10 '13 at 01:20

user2849499

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3 answers

Speed of object towards a point not in the object's trajectory?

Trying to study for my mid-term, but I'm having slight difficulties understanding what I'm supposed to do in this one problem: A batter starts running towards first base at a constant speed of 6 m/s. The distance between each adjacent plate is 27.5…

derivatives

asked Oct 09 '13 at 21:18

Netfangled

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1 answer

The derivative of a trigonometric function.

Let: $$f(x) = \sin x + \frac{\sin3x}3 + \frac{\sin5x}5 + \frac{\sin7x}7$$ Prove that: $$f' \left(\frac{\pi}{9}\right) = \frac{1}{2}$$ Could somebody point me in the right direction with this one?

derivatives

asked Oct 01 '13 at 18:48