MENUFirst Derivative Calculator(Solver) with Steps
Free derivatives calculator(solver) that gets the detailed solution of the first derivative of a function.
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Let [tex]u = \frac{1}{\cos{\left (6 w \right )}}[/tex].
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The derivative of sine is cosine:
[tex]\frac{d}{d u} \sin{\left (u \right )} = \cos{\left (u \right )}[/tex]
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Then, apply the chain rule. Multiply by [tex]\frac{d}{d w} \frac{1}{\cos{\left (6 w \right )}}[/tex]:
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Let [tex]u = \cos{\left (6 w \right )}[/tex].
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Apply the power rule: [tex]\frac{1}{u}[/tex] goes to [tex]- \frac{1}{u^{2}}[/tex]
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Then, apply the chain rule. Multiply by [tex]\frac{d}{d w} \cos{\left (6 w \right )}[/tex]:
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Let [tex]u = 6 w[/tex].
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The derivative of cosine is negative sine:
[tex]\frac{d}{d u} \cos{\left (u \right )} = - \sin{\left (u \right )}[/tex]
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Then, apply the chain rule. Multiply by [tex]\frac{d}{d w}\left(6 w\right)[/tex]:
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The derivative of a constant times a function is the constant times the derivative of the function.
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Apply the power rule: [tex]w[/tex] goes to [tex]1[/tex]
So, the result is: [tex]6[/tex]
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The result of the chain rule is:
[tex]- 6 \sin{\left (6 w \right )}[/tex]
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The result of the chain rule is:
[tex]\frac{6 \sin{\left (6 w \right )}}{\cos^{2}{\left (6 w \right )}}[/tex]
The result of the chain rule is:
[tex]\frac{6 \cos{\left (\frac{1}{\cos{\left (6 w \right )}} \right )}}{\cos^{2}{\left (6 w \right )}} \sin{\left (6 w \right )}[/tex]
The answer is:
[tex]\frac{6 \cos{\left (\frac{1}{\cos{\left (6 w \right )}} \right )}}{\cos^{2}{\left (6 w \right )}} \sin{\left (6 w \right )}[/tex]
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* is multiplication
oo is $\infty$
pi is $\pi$
x^2 is x2
sqrt(x) is $\sqrt{x}$
sqrt[3](x) is $\sqrt[3]{x}$
(a+b)/(c+d) is $\frac{a+b}{c+d}$
The Most Important Derivatives - Basic Formulas/Rules
$\frac{d}{dx}a=0$ (a is a constant)
$\frac{d}{dx}x=1$
$\frac{d}{dx}x^n=nx^{n-1}$
$\frac{d}{dx}e^x=e^x$
$\frac{d}{dx}\log x=\frac1x$
$\frac{d}{dx}a^x=a^x\log x$
$(f\ g)' = f'g + fg'$ - Product Rule
$(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$ - Quotient Rule
$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$ - Chain Rule
$\frac{d}{dx}\sin(x)=\cos(x)$
$\frac{d}{dx}\cos(x)=-\sin(x)$
$\frac{d}{dx}\tan(x)=\sec^2(x)$
$\frac{d}{dx}\cot(x)=-csc^2(x)$
$\frac{d}{dx}\arcsin(x)=\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\arccos(x)=-\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\arctan(x)=\frac{1}{1+x^2}$
$\frac{d}{dx}\text{arccot}(x)=-\frac{1}{1+x^2}$
$\frac{d}{dx}\text{arcsec}(x)=\frac{1}{x\sqrt{x^2-1}}$
$\frac{d}{dx}\text{arccsc}(x)=-\frac{1}{x\sqrt{x^2-1}}$
