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Popular >

derivative of y^x

Solution

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

Solution

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

Solution steps

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

Apply exponent rule:

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

y^{x} = e^{x l n (y)}

= \frac{d}{d x} (e^{x l n (y)})

Apply the chain rule:

e^{x l n (y)} \frac{d}{d x} (x ln (y))

= e^{x l n (y)} \frac{d}{d x} (x ln (y))

\frac{d}{d x} (x ln (y)) = ln (y) + \frac{x \frac{d y}{d x}}{y}

= e^{x l n (y)} (ln (y) + \frac{x \frac{d y}{d x}}{y})

e^{x l n (y)} = y^{x}

= (ln (y) + \frac{x \frac{d y}{d x}}{y}) y^{x}

Popular Examples

derivative of f(x)=7x+6 derivative of

f (x) = 7 x + 6

integral of 1/(2sqrt(tan(x)))

\int \frac{1}{2 tan ( x )} d x

inverse oflaplace (s^2+3s+5)/(s^3(s+2))inverse laplace

\frac{s ^{2} + 3 s + 5}{s ^{3} ( s + 2 )}

(dy)/(dx)=sqrt(x)

\frac{d y}{d x} = x

sum from n=0 to infinity of 2/(n^2)

n = 0 \sum \infty \frac{2}{n ^{2}}

Frequently Asked Questions (FAQ)

What is the derivative of y^x ?
The derivative of y^x is (ln(y)+(x(dy)/(dx))/y)y^x
What is the first derivative of y^x ?
The first derivative of y^x is (ln(y)+(x(dy)/(dx))/y)y^x