\(\dfrac{\mathrm d}{\mathrm dx}\mathrm e^x=\mathrm e^x\qquad\dfrac{\mathrm d}{\mathrm dx}\mathrm e^{

Question

$\dfrac{\mathrm d}{\mathrm dx}\mathrm e^x=\mathrm e^x\qquad\dfrac{\mathrm d}{\mathrm dx}\mathrm e^{\mathrm{f}(x)}=\mathrm{f}^{\prime}(x)\mathrm e^{\mathrm{f}(x)}$ Differentiate the exponent and multiply by the original exponential function. Example Differentiate $\mathrm e^{7x^2+1}$ with respect to $x$. $\dfrac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^{7x^2+1}$${}=\mathrm{e}^{7x^2+1}(14x)$${}=14x\mathrm{e}^{7x^2+1}$ Exam check: the power does not drop: $\dfrac{\mathrm d}{\mathrm dx}\mathrm e^{3x}=3\mathrm e^{3x}$, not $3x\mathrm e^{3x-1}$. The power rule applies only when the exponent is a constant.

Accepted Answer

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm e^x=\mathrm e^x$, $\frac{\mathrm{d}}{\mathrm{d}x}\mathrm e^{7x^2+1}=14x\mathrm e^{7x^2+1}$

Basics of Exponential Derivative

Basics of Exponential Derivative

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