What is the derivative of 4^{9^{x^2}} ?

The derivative of 4^{9^{x^2}} is ln(2)ln(3)*9^{x^2}*2^{2*9^{x^2}+3}x

What is the first derivative of 4^{9^{x^2}} ?

The first derivative of 4^{9^{x^2}} is ln(2)ln(3)*9^{x^2}*2^{2*9^{x^2}+3}x

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\frac{d}{d x} (4^{9^{x^{2}}})

ln (2) ln (3) \cdot 9^{x^{2}} \cdot 2^{2 \cdot 9^{x^{2}} + 3} x

Solution steps

\frac{d}{d x} (4^{9^{x^{2}}})

Apply exponent rule:

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

4^{9^{x^{2}}} = e^{9^{x^{2}} l n (4)}

= \frac{d}{d x} (e^{9^{x^{2}} l n (4)})

Apply the chain rule:

e^{9^{x^{2}} l n (4)} \frac{d}{d x} (9^{x^{2}} ln (4))

= e^{9^{x^{2}} l n (4)} \frac{d}{d x} (9^{x^{2}} ln (4))

\frac{d}{d x} (9^{x^{2}} ln (4)) = 8 ln (2) ln (3) \cdot 9^{x^{2}} x

= e^{9^{x^{2}} l n (4)} \cdot 8 ln (2) ln (3) \cdot 9^{x^{2}} x

Simplify

e^{9^{x^{2}} l n (4)} \cdot 8 ln (2) ln (3) \cdot 9^{x^{2}} x : ln (2) ln (3) \cdot 9^{x^{2}} \cdot 2^{2 \cdot 9^{x^{2}} + 3} x

= ln (2) ln (3) \cdot 9^{x^{2}} \cdot 2^{2 \cdot 9^{x^{2}} + 3} x

What is the derivative of 4^{9^{x^2}} ?
The derivative of 4^{9^{x^2}} is ln(2)ln(3)*9^{x^2}*2^{2*9^{x^2}+3}x
What is the first derivative of 4^{9^{x^2}} ?
The first derivative of 4^{9^{x^2}} is ln(2)ln(3)*9^{x^2}*2^{2*9^{x^2}+3}x