What is the derivative of 5^{tan(5x}) ?

The derivative of 5^{tan(5x}) is ln(5)*5^{tan(5x)+1}sec^2(5x)

What is the first derivative of 5^{tan(5x}) ?

The first derivative of 5^{tan(5x}) is ln(5)*5^{tan(5x)+1}sec^2(5x)

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\frac{d}{d x} (5^{t a n (5 x)})

ln (5) \cdot 5^{t a n (5 x) + 1} sec^{2} (5 x)

Solution steps

\frac{d}{d x} (5^{t a n (5 x)})

Apply exponent rule:

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

5^{t a n (5 x)} = e^{t a n (5 x) l n (5)}

= \frac{d}{d x} (e^{t a n (5 x) l n (5)})

Apply the chain rule:

e^{t a n (5 x) l n (5)} \frac{d}{d x} (tan (5 x) ln (5))

= e^{t a n (5 x) l n (5)} \frac{d}{d x} (tan (5 x) ln (5))

\frac{d}{d x} (tan (5 x) ln (5)) = ln (5) sec^{2} (5 x) \cdot 5

= e^{t a n (5 x) l n (5)} ln (5) sec^{2} (5 x) \cdot 5

Simplify

e^{t a n (5 x) l n (5)} ln (5) sec^{2} (5 x) \cdot 5 : ln (5) \cdot 5^{t a n (5 x) + 1} sec^{2} (5 x)

= ln (5) \cdot 5^{t a n (5 x) + 1} sec^{2} (5 x)

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What is the derivative of 5^{tan(5x}) ?
The derivative of 5^{tan(5x}) is ln(5)*5^{tan(5x)+1}sec^2(5x)
What is the first derivative of 5^{tan(5x}) ?
The first derivative of 5^{tan(5x}) is ln(5)*5^{tan(5x)+1}sec^2(5x)