What is the derivative of 1/(f(x^2(x))) ?

The derivative of 1/(f(x^2(x))) is -(3{f}(x)+x derivative of {f}(x))/(x^4{f)(x)^2}

What is the first derivative of 1/(f(x^2(x))) ?

The first derivative of 1/(f(x^2(x))) is -(3{f}(x)+x derivative of {f}(x))/(x^4{f)(x)^2}

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\frac{d}{d x} (\frac{1}{f ( x ) ( x ) ^{2} ( x )})

- \frac{3 f ( x ) + x \frac{d}{d x} ( f ( x ) )}{x ^{4} f ( x ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{1}{f ( x ) ( x ) ^{2} ( x )})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \frac{d}{d x} ((f (x) x^{2} x)^{- 1})

Apply the chain rule:

- \frac{1}{( f ( x ) x ^{2} x ) ^{2}} \frac{d}{d x} (f (x) x^{2} x)

= - \frac{1}{( f ( x ) x ^{2} x ) ^{2}} \frac{d}{d x} (f (x) x^{2} x)

\frac{d}{d x} (f (x) x^{2} x) = 3 x^{2} f (x) + \frac{d}{d x} (f (x)) x^{3}

= - \frac{1}{( f ( x ) x ^{2} x ) ^{2}} (3 x^{2} f (x) + \frac{d}{d x} (f (x)) x^{3})

Simplify

- \frac{1}{( f ( x ) x ^{2} x ) ^{2}} (3 x^{2} f (x) + \frac{d}{d x} (f (x)) x^{3}) : - \frac{3 f ( x ) + x \frac{d}{d x} ( f ( x ) )}{x ^{4} f ( x ) ^{2}}

= - \frac{3 f ( x ) + x \frac{d}{d x} ( f ( x ) )}{x ^{4} f ( x ) ^{2}}

- 10 x^{3} + 6 x^{40} y + y^{4} = 3

\int \frac{x ^{3}}{1 - x ^{2}} d x

x \to 4 - lim (2 - x - x^{2})

f (x) = (x^{3} + 9) e^{x}

\int_{e}^{e^{3}} \frac{15}{y} d y

What is the derivative of 1/(f(x^2(x))) ?
The derivative of 1/(f(x^2(x))) is -(3{f}(x)+x derivative of {f}(x))/(x^4{f)(x)^2}
What is the first derivative of 1/(f(x^2(x))) ?
The first derivative of 1/(f(x^2(x))) is -(3{f}(x)+x derivative of {f}(x))/(x^4{f)(x)^2}