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Basic Derivatives, Product Rule, Quotient Rule, Chain Rule, Implicit Differentiation, Differentiating trigonometric and logarithmic functions, tangents

rebel

rebelmaths

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$f(x) = \\var{a}$

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$f(x) = \\var{b}x^2 -\\var{c}x+\\var{d}$

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$f(x) = x^2(1-\\var{f}x)$

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$f(x) = -\\frac{\\var{g}}{x^\\var{h}}$

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$f(x) = \\sqrt{x}$

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$f(x) = x^2 + e^x$

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Differentiate the following with respect to $x$.

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Some basic derivatives.

rebelmaths

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Find the equation of the tangent line to the curve $f(x) = \\var{a}x^2 - \\var{b}x$ at the point $(\\var{c},\\simplify{{a}{c}{c}-{b}{c}})$.

$y = $[[0]]

Find an equation of the tangent line to the curve $f(x) = \\sqrt[4]{x}$ at the point $(1,1)$.

$y = $ [[0]]

Find an equation of the tangent line to $y = x^4 + \\var{d}x^2-\\var{f}x$ at the point $(1,\\simplify{1+{d}-{f}})$.

$y = $ [[0]]

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rebelmaths

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The velocity as a function of $t$ is

$v =$ [[0]]ms$^{-1}$

The acceleration as a function of $t$ is

$a = $ [[0]]ms$^{-2}$.

The acceleration after 2s is: [[0]]ms$^{-2}$

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The acceleration when the velocity is 0 is: [[0]]ms$^{-2}$.

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The equation of motion of a particle is $s = t^\\var{a} - \\var{a}t$, where $s$ is in metres and $t$ is in seconds.

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$(x^\\var{a}+\\var{b}x)e^x$.

(In your answer if writing $xe^x$ be sure to write it as x*e^x)

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$\\frac{\\var{c}x-\\var{d}}{\\var{f}x+\\var{g}}$

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$(u-\\sqrt{u})(u+\\sqrt{u})$

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$(\\frac{\\var{h}}{y^2}-\\frac{\\var{j}}{y^4})(y+\\var{k}y^3)$

Answer: [[0]] $+ \\frac{1}{y^2}$[[1]] $+ \\frac{1}{y^4}$[[2]].

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Differentiate the following using the product and quotient rules where appropriate.

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Differentiate $\\var{a}x^2-\\var{b}\\cos(x)$.

When writing your answer be sure to include the brackets in any trigonometric expressions, i.e. write sin(x) rather than sinx.

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Differentiate $\\frac{\\cos(x)}{1-\\sin{x}}$.

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Find an equation of the tangent line to the curve $y = e^x\\cos(x)$ at the point $(0,1)$.

$y = $ [[0]]

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$(x^4 + \\var{a}x^2 -\\var{b})^\\var{c}$

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$\\cos(x^3)$

Take care here to write the trigonometric expressions with brackets, i.e. use cos(x) rather than cosx.

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$xe^{-\\var{d}x}$

In your answer if you want to write something in the form $xe^x$ be sure to include the symbol * as in x*e^x.

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Differentiate the following with respect to $x$.

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$xy+\\var{a}x+\\var{b}x^2=\\var{c}$.

$\\frac{dy}{dx} = $ [[0]]

$x^\\var{d}+y^\\var{d}=1$

$\\frac{dy}{dx} = $[[0]]

$xe^y=x-y$

$\\frac{dy}{dx} = $[[0]].

When writing $xe^y$ in your answer be sure to include a * symbol for multiplication i.e. x*e^y.

$x\\sin(y) + y\\sin(x) = 1$

$\\frac{dy}{dx} = $ [[0]].

When writing $x\\cos(y)$ in your answer be sure to include a * symbol for multiplication i.e. x*cos(y).

Find $\\frac{dy}{dx}$ using implicit differentiation in the following:

Differentiate $x\\ln(x)-x$

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Differentiate $ln(x^\\var{a}+1)$

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Differentiate $\\ln(\\ln(x))$

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$y = x^2\\ln(2x)$

$y'(x) = $ [[0]].

$y''(x) = $ [[1]].

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