What is the derivative of (sin^2(x+1)^x) ?

The derivative of (sin^2(x+1)^x) is (ln(sin^2(x)+1)+(xsin(2x))/(sin^2(x)+1))(sin^2(x)+1)^x

What is the first derivative of (sin^2(x+1)^x) ?

The first derivative of (sin^2(x+1)^x) is (ln(sin^2(x)+1)+(xsin(2x))/(sin^2(x)+1))(sin^2(x)+1)^x

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

(ln (sin^{2} (x) + 1) + \frac{x sin ( 2 x )}{sin ^{2} ( x ) + 1}) (sin^{2} (x) + 1)^{x}

Solution steps

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

Apply exponent rule:

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

(sin^{2} (x) + 1)^{x} = e^{x l n (s i n^{2} (x) + 1)}

= \frac{d}{d x} (e^{x l n (s i n^{2} (x) + 1)})

Apply the chain rule:

e^{x l n (s i n^{2} (x) + 1)} \frac{d}{d x} (x ln (sin^{2} (x) + 1))

= e^{x l n (s i n^{2} (x) + 1)} \frac{d}{d x} (x ln (sin^{2} (x) + 1))

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

= e^{x l n (s i n^{2} (x) + 1)} (ln (sin^{2} (x) + 1) + \frac{x sin ( 2 x )}{sin ^{2} ( x ) + 1})

e^{x l n (s i n^{2} (x) + 1)} = (sin^{2} (x) + 1)^{x}

= (ln (sin^{2} (x) + 1) + \frac{x sin ( 2 x )}{sin ^{2} ( x ) + 1}) (sin^{2} (x) + 1)^{x}

4 t

\frac{d}{d x} (sin^{3} (3 x - 1))

e^{x}, 5

x \to 3 lim (\frac{∣ x - 1 ∣}{x - 1})

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

What is the derivative of (sin^2(x+1)^x) ?
The derivative of (sin^2(x+1)^x) is (ln(sin^2(x)+1)+(xsin(2x))/(sin^2(x)+1))(sin^2(x)+1)^x
What is the first derivative of (sin^2(x+1)^x) ?
The first derivative of (sin^2(x+1)^x) is (ln(sin^2(x)+1)+(xsin(2x))/(sin^2(x)+1))(sin^2(x)+1)^x