What is the derivative of ln(x^5(x+4^7(x^2+4)^5)) ?

The derivative of ln(x^5(x+4^7(x^2+4)^5)) is (2(11x^3+30x^2+24x+40))/(x(x+4)(x^2+4))

What is the first derivative of ln(x^5(x+4^7(x^2+4)^5)) ?

The first derivative of ln(x^5(x+4^7(x^2+4)^5)) is (2(11x^3+30x^2+24x+40))/(x(x+4)(x^2+4))

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\frac{d}{d x} (ln (x^{5} (x + 4)^{7} (x^{2} + 4)^{5}))

\frac{2 ( 11 x ^{3} + 30 x ^{2} + 24 x + 40 )}{x ( x + 4 ) ( x ^{2} + 4 )}

Solution steps

\frac{d}{d x} (ln (x^{5} (x + 4)^{7} (x^{2} + 4)^{5}))

Apply the chain rule:

\frac{1}{x ^{5} ( x + 4 ) ^{7} ( x ^{2} + 4 ) ^{5}} \frac{d}{d x} (x^{5} (x + 4)^{7} (x^{2} + 4)^{5})

= \frac{1}{x ^{5} ( x + 4 ) ^{7} ( x ^{2} + 4 ) ^{5}} \frac{d}{d x} (x^{5} (x + 4)^{7} (x^{2} + 4)^{5})

\frac{d}{d x} (x^{5} (x + 4)^{7} (x^{2} + 4)^{5}) = 5 x^{4} (x + 4)^{7} (x^{2} + 4)^{5} + (7 (x + 4)^{6} (x^{2} + 4)^{5} + 10 x (x^{2} + 4)^{4} (x + 4)^{7}) x^{5}

= \frac{1}{x ^{5} ( x + 4 ) ^{7} ( x ^{2} + 4 ) ^{5}} (5 x^{4} (x + 4)^{7} (x^{2} + 4)^{5} + (7 (x + 4)^{6} (x^{2} + 4)^{5} + 10 x (x^{2} + 4)^{4} (x + 4)^{7}) x^{5})

Simplify

\frac{1}{x ^{5} ( x + 4 ) ^{7} ( x ^{2} + 4 ) ^{5}} (5 x^{4} (x + 4)^{7} (x^{2} + 4)^{5} + (7 (x + 4)^{6} (x^{2} + 4)^{5} + 10 x (x^{2} + 4)^{4} (x + 4)^{7}) x^{5}) : \frac{2 ( 11 x ^{3} + 30 x ^{2} + 24 x + 40 )}{x ( x + 4 ) ( x ^{2} + 4 )}

= \frac{2 ( 11 x ^{3} + 30 x ^{2} + 24 x + 40 )}{x ( x + 4 ) ( x ^{2} + 4 )}

\int_{0}^{4} x + 2 - x d x

\int \frac{y ^{2} + 16}{y} d y

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

f (x) = \frac{2 x + 1}{x + 2}

\frac{d}{d x} (\frac{x}{2})

What is the derivative of ln(x^5(x+4^7(x^2+4)^5)) ?
The derivative of ln(x^5(x+4^7(x^2+4)^5)) is (2(11x^3+30x^2+24x+40))/(x(x+4)(x^2+4))
What is the first derivative of ln(x^5(x+4^7(x^2+4)^5)) ?
The first derivative of ln(x^5(x+4^7(x^2+4)^5)) is (2(11x^3+30x^2+24x+40))/(x(x+4)(x^2+4))