What is the derivative of ({f}(x(x^3))^2) ?

The derivative of ({f}(x(x^3))^2) is 6x^5{f}(x)^2+2{f}(x) derivative of {f}(x)x^6

What is the first derivative of ({f}(x(x^3))^2) ?

The first derivative of ({f}(x(x^3))^2) is 6x^5{f}(x)^2+2{f}(x) derivative of {f}(x)x^6

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

6 x^{5} f (x)^{2} + 2 f (x) \frac{d}{d x} (f (x)) x^{6}

Solution steps

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

Simplify

(f (x) (x^{3}))^{2} : x^{6} f (x)^{2}

= \frac{d}{d x} (x^{6} f (x)^{2})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{6}, g = f (x)^{2}

= \frac{d}{d x} (x^{6}) f (x)^{2} + \frac{d}{d x} (f (x)^{2}) x^{6}

\frac{d}{d x} (x^{6}) = 6 x^{5}

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

= 6 x^{5} f (x)^{2} + 2 f (x) \frac{d}{d x} (f (x)) x^{6}

\frac{d}{d x} (3 cot (2 x) sec (2 x))

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

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

\frac{d}{d x} (e^{x^{2} + x + 1})

\frac{6}{( 1 - x ) ^{3}}

What is the derivative of ({f}(x(x^3))^2) ?
The derivative of ({f}(x(x^3))^2) is 6x^5{f}(x)^2+2{f}(x) derivative of {f}(x)x^6
What is the first derivative of ({f}(x(x^3))^2) ?
The first derivative of ({f}(x(x^3))^2) is 6x^5{f}(x)^2+2{f}(x) derivative of {f}(x)x^6