What is the derivative of (1/(x^2)^{tan(x)}) ?

The derivative of (1/(x^2)^{tan(x)}) is (1/(x^2))^{tan(x)}(-2sec^2(x)ln(x)-(2tan(x))/x)

What is the first derivative of (1/(x^2)^{tan(x)}) ?

The first derivative of (1/(x^2)^{tan(x)}) is (1/(x^2))^{tan(x)}(-2sec^2(x)ln(x)-(2tan(x))/x)

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\frac{d}{d x} ((\frac{1}{x ^{2}})^{t a n (x)})

(\frac{1}{x ^{2}})^{t a n (x)} (- 2 sec^{2} (x) ln (x) - \frac{2 tan ( x )}{x})

Solution steps

\frac{d}{d x} ((\frac{1}{x ^{2}})^{t a n (x)})

Apply exponent rule:

a^{b} = e^{b l n (a)}

(\frac{1}{x ^{2}})^{t a n (x)} = e^{t a n (x) l n (\frac{1}{x ^{2}})}

= \frac{d}{d x} (e^{t a n (x) l n (\frac{1}{x ^{2}})})

Apply the chain rule:

e^{t a n (x) l n (\frac{1}{x ^{2}})} \frac{d}{d x} (tan (x) ln (\frac{1}{x ^{2}}))

= e^{t a n (x) l n (\frac{1}{x ^{2}})} \frac{d}{d x} (tan (x) ln (\frac{1}{x ^{2}}))

\frac{d}{d x} (tan (x) ln (\frac{1}{x ^{2}})) = - sec^{2} (x) ln (x^{2}) - \frac{2 tan ( x )}{x}

= e^{t a n (x) l n (\frac{1}{x ^{2}})} (- sec^{2} (x) ln (x^{2}) - \frac{2 tan ( x )}{x})

Simplify

e^{t a n (x) l n (\frac{1}{x ^{2}})} (- sec^{2} (x) ln (x^{2}) - \frac{2 tan ( x )}{x}) : (\frac{1}{x ^{2}})^{t a n (x)} (- 2 sec^{2} (x) ln (x) - \frac{2 tan ( x )}{x})

= (\frac{1}{x ^{2}})^{t a n (x)} (- 2 sec^{2} (x) ln (x) - \frac{2 tan ( x )}{x})

\frac{3}{( 4 x + 1 )}

y^{^{'}} + (\frac{2}{t}) y = \frac{cos ( t )}{t ^{2}}

\frac{\partial}{\partial x} (4 x^{8} y^{6} + 6 x^{7} y^{5})

x + 2^{x}

9 e^{x^{2}}

What is the derivative of (1/(x^2)^{tan(x)}) ?
The derivative of (1/(x^2)^{tan(x)}) is (1/(x^2))^{tan(x)}(-2sec^2(x)ln(x)-(2tan(x))/x)
What is the first derivative of (1/(x^2)^{tan(x)}) ?
The first derivative of (1/(x^2)^{tan(x)}) is (1/(x^2))^{tan(x)}(-2sec^2(x)ln(x)-(2tan(x))/x)